Talk:Centrifugal force/Archive 13

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Absolute rotation induces an absolute outward expansion pressure

This statement by David seems to encapsulate his viewpoint. I guess one can think of rotation without specifying relative to what. A question arises though: who gets to decide how rapid the rotation is? Can we identify the stationary observer who actually has got the rotation correct? How is that done?

If we cannot do that, everybody has a legitimate view of how fast the rotation is, and not everyone will agree. So, by implication, not everyone will agree on the "absolute outward pressure", which then is not absolute.

Newton solved this problem: if your calculation using Newton's laws with the rate you observe works to describe all the observations while using zero "absolute outward pressure", then you are in a stationary frame. In other words, the absolute "outward expansion pressure" is identically zero.

How do you describe the situation, David? Brews ohare (talk) 00:37, 22 May 2009 (UTC)

Brews, You have indeed asked the ultimate relevant question. That is what the rotating bucket experiment is all about. It shows that centrifugal force is a real outward expansion pressure that is induced as a result of absolute rotation relative to one frame of reference in particular. That frame of reference is marked out by the background stars.

Those are the facts. But it seems that alot of the problem here is that people trained in relativity think that this is not fair because it gives special physical significance to one frame of reference in particular. My guess is that here lies the root of the entire edit war. These people are trying to put their own modern rationalization on something which doesn't fit with how they think that things should be. And so they are trying to suppress the simplicity of the topic. David Tombe (talk) 11:49, 22 May 2009 (UTC)

It would appear that Newton suggests that absolute rotation is wrt the fixed stars. However, if the observer is stationary wrt the fixed stars, Newton says there is zero absolute outward pressure. Establishing this point is specifically his objective with the bucket argument and with the rotating spheres. So, my conclusion is that you disagree with Newton about this, or interpret his objective in these two examples as something different? Brews ohare (talk) 12:21, 22 May 2009 (UTC)

Brews, on this issue, I totally agree with Newton. The frame of reference that is marked out by the background stars, possesses special properties which become manifest when rotation occurs relative to any arbitrarily chosen point in space, relative to the background stars. But that doesn't mean that I relate the stars themselves directly to the physical cause. That's another matter and I'd be going into original research. I noticed that you have been editing on the electric dipole page. You will have seen that the dipole field is an inverse cube law force field. Does that not give you a clue?

I do however disagree with Newton's attempt to denigrate Leibniz's equation. Clearly Leibniz beat Newton to it. Leibniz got both the inverse square law for gravity and the inverse cube law for centrifugal force. He put it all into one equation. He was ahead of Newton. Newton only got as far as the inverse square law for gravity. Newton sabotaged Leibniz's equation by mixing it all up with his own third law of action and reaction. Hence, Newton's reactive centrifugal force is wrong in principle, but it is good in practice for circular motion scenarios.David Tombe (talk) 13:15, 22 May 2009 (UTC)

David: Your reply is non-responsive. If the observer is stationary wrt the fixed stars, Newton says there is zero absolute outward pressure. Establishing this point is specifically his objective with the bucket argument and with the rotating spheres. So, my conclusion is that you disagree with Newton about this, or interpret his objective in these two examples as something different? Brews ohare (talk) 13:23, 22 May 2009 (UTC)

Brews, I don't get your point. If it's rotating realtive to the background stars, there will be an outward expansion pressure. If it's not rotating relative to the background stars, there won't be. I assume that Newton was saying this too. David Tombe (talk) 13:27, 22 May 2009 (UTC)

I don't think Newton agrees with you. For example, with the rotating spheres Newton's suggestion is that a string tension is required in the frame stationary wrt the fixed stars only to provide centripetal force. The centrifugal force is zero in this frame. Brews ohare (talk) 16:49, 22 May 2009 (UTC)

No Brews, The centrifugal force on an object, relative to a point, depends on that object's own transverse speed relative to the background stars. In your rotating spheres example, the centrifugal force comes from their own transverse speed relative to the inertial frame (marked out by the background stars). That transverse speed induces an outward expansion which pulls the string taut. The tension in the taut string then applies an inward centripetal force which makes the spheres move in circular motion. In the circular motion, the inward centripetal force is equal and opposite to the outward centrifugal force. David Tombe (talk) 18:59, 22 May 2009 (UTC)

David: In the frame stationary wrt the fixed stars, the taut string provides a centripetal force that provides the inward acceleration necessary to change the direction of the velocity so it remains tangent to the circle, so circular motion occurs. There is no balancing of two forces, as that leads to zero acceleration and therefore to straight-line motion. I have a feeling of deja vu here. Brews ohare (talk) 23:05, 22 May 2009 (UTC)

Brews, you are falling into the trap of ignoring the outward centrifugal force that exists relative to the chosen origin, even when the object is moving in a fly-by straight line prior to the application of any centripetal force. That is the bit that modern physics teachers have overlooked. But it is not overlooked in the planetary orbital equation. According to the radial planetary orbital equation (Leibniz, or 3-12 in Goldstein, or problem 8-23 in Taylor), when circular motion occurs, r double dot will be zero, and the inward centripetal force will be exactly balanced by an outward inverse cube law centrifugal force. If there is no centripetal force, the centrifugal force alone will cause a hyperbolic orbit of infinite eccentricity, which is a straight line fly-by motion.

I have discussed this on forums. Alot of people can't see where the outward pressure actually lies in a fly-by straight line motion. But mathematically speaking, the centrifugal force is nevertheless there. It is built into the geometry of space. You can see it better if you extrapolate the situation to a four-body problem involving two adjacent two body orbits. If we criss-cross centrifugal force over any pair within the four, the two orbits will repel each other. That is how Maxwell explained magnetic repulsion. David Tombe (talk) 10:27, 23 May 2009 (UTC)

If that worked, it would be incredibly trivial to demonstrate, just spin a couple of objects next to each other and you would get a massive repulsion. That doesn't happen. It's a ridiculous misunderstanding of the physics.- (User) Wolfkeeper (Talk) 12:56, 29 May 2009 (UTC)

No Wolfkeeper, you are not looking at the whole picture. The context was Maxwell's sea of molecular vortices and the centrifugal repulsion is between the tiny vortices. Two large spinning objects immersed in such a sea would not repel each other because the effect would be absorbed by the sea of vortices. There would be no head on centrifugal repulsion between the two large objects. David Tombe (talk) 15:22, 29 May 2009 (UTC)

We're not talking about electromagnetism we're talking about simple mechanics. If the (non reactive) centrifugal force is a real force rather than being an artifact of the coordinate or reference frame (a 'pseudo force') then you would get a real repulsive force. Your raising of E-M theory to do with the curl of the magnetic field is a non sequitor; there is not magnetic field present, nor was there in the gravitational case you were referring to either. The true explanation is that the non reactive centrifugal force in polar coordinate systems is a pseudoforce/frame acceleration, and hence becomes simple inertia in non rotating frames of reference. It's that simple.- (User) Wolfkeeper (Talk) 16:06, 29 May 2009 (UTC)

Wolfkeeper, ultimately centrifugal force is an electromagnetic effect. It lies central place in EM theory and appears as the term vXB. You can take it or leave it. Here is not the place to discuss these matters. I can only draw your attention to ideas such as extrapolating Leibniz's equation to the four body problem of two dipoles, and to Maxwell's 1861 paper. In the end, we'll have to agree to disagree. Your blinkered view stems from the fact that in everyday life, centrifugal force is only experienced passively. David Tombe (talk) 16:21, 29 May 2009 (UTC)

Um... no. Centrifugal force in polar coordinates or rotating reference frames is not an electromagnetic effect it is inertia. Unless you're claiming that inertia is also an electromagnetic effect, in which case you would have to: a) prove it and then b) collect your Nobel prize. In the meantime I'm going to: c) laugh at you; this is the most ridiculous thing I have ever heard.- (User) Wolfkeeper (Talk) 17:13, 29 May 2009 (UTC)

Yes of course inertia is an electromagnetic effect. It's about accumulated pressure in the same medium in which light propagates. David Tombe (talk) 20:15, 29 May 2009 (UTC)

LOL. Reference? (LOL)- (User) Wolfkeeper (Talk) 22:34, 29 May 2009 (UTC)

David: I despair of reaching any common ground with you as long as you stick to the planetary problem. I suggest that the focus be changed to the case of two identical spheres tied by a cord, and get the connection to gravity and Kepler's laws out of the picture. That puts things in a clearer context without unnecessary side issues. Within the rotating spheres problem, I agree with the presentations at rotating spheres. Maybe you would like to discuss centrifugal force in that case? Brews ohare (talk) 17:26, 23 May 2009 (UTC)

Brews, Planetary orbital theory is not a side issue. The equation,

${\ddot {r}}=centripetalforce+l^{2}/r^{3}$

or,

${\ddot {r}}=centripetalforce+centrifugalforce$

covers for any scenario in this topic that you could possibly imagine. When the centripetal force is gravity, it becomes Leibniz's equation. So let's look at your rotating spheres. Let's start with the situation before the string is attached. If we ignore gravity, the two spheres will move with a mutual transverse speed and they will both move in a straight line. This is the infinitely eccentric hyperbolic fly-by solution to the equation above, which arises when the centripetal force is zero. The line joining the two will be rotating, and there will be an outward centrifugal force acting along that line which is proportional to the inverse cube of the distance between the two spheres.

If we then attach a string, the centrifugal force will pull the string taut. The tension in the taut string will then cause an inward centripetal force to act. We will then have a circular motion scenario in which the outward centrifugal force is exactly balanced by an inward centripetal force, and as such ${\ddot {r}}$ will be equal to zero.

Remember this golden rule,

(1) Rotation causes centrifugal force

(2) Centripetal force causes curved path motion

and you will not go wrong. David Tombe (talk) 18:39, 23 May 2009 (UTC)

The equation l^2/r^3 is not physically different from rw^2, you just substitute l=r^2 w into the equation and you get the same thing. It's only trivially different.- (User) Wolfkeeper (Talk) 00:29, 24 May 2009 (UTC)

Wolfkeeper, You are absolutely correct, and I was fully aware of that fact. Do remember however that the inverse cube law form only holds in the special case when angular momentum is conserved. If you are genuinely interested in this topic, then there is hope that we might be able to make progress. Your edit above tells me that you have just realized something which I have been trying to draw your attention to for a long time. David Tombe (talk) 12:45, 24 May 2009 (UTC)

David: Your equations are tied to polar coordinates, and more than that, to a specific terminology attached to each term. That terminology prejudges the issues. To clarify what the forces and vectors are, one way is to leave polar coordinates and return to a formulation in a coordinate-independent language using vector calculus. In such language, d²r / dt² is acceleration. This acceleration in uniform circular motion is supplied by a vector force applied to the body due to the tension in the string all by itself, and no centrifugal force vector appears in the equations. On the other hand, in polar coordinates, translation of d²r / dt² introduces not only the d²r / dt² term, but another term you call the centrifugal force, but which is an artifact of the coordinate choice. Clearly, the translation of d²r / dt² into the language of a particular coordinate system does not reflect the nature of the problem being solved or the physical forces at work in a particular problem; rather it is a translation to the chosen coordinate system that is independent of the physical problem and dependent on the choice of coordinates. If you chose instead parabolic coordinates, this extra term would be different, again reflecting that it originates in the choice of coordinates, not in the forces at play. Brews ohare (talk) 20:52, 23 May 2009 (UTC)

Brews, we can express the equation in plane English without using any mathematical symbolism at all. The physics is not dependent on the language that is used to describe it. The single equation which applies to any scenario which you could possibly present me with in this topic is,

Radial Force = Centripetal Force + Centrifugal force

You give me any scenario that you like and I will analyze it for you in terms of this equation. Leibniz's equation is simply a more mathematically detailed version of this equation which specifically uses the inverse square law of gravity for the centripetal force along with the inverse cube law version of centrifugal force, since angular momentum is conserved in planetary orbits.

I think that you need to open your eyes a bit wider to the true physical nature of centrifugal force. In the scenarios which you are considering, we only ever feel it passively. We can see it, but we can only ever feel it when a centripetal force opposes it. That's part of what makes some people erroneously state that it is fictitious, or that it is only a reaction to a centripetal force. But there are a few examples in nature where we can feel an active centrifugal force head on in its pure state. One such example is when we push two like magnetic poles together. You know what that feels like. That is pure centrifugal force felt directly. Another situation where you can get that exact same feeling is when you force precess a spinning gyroscope and restrain it to the rotation plane that you are force precessing it in. If you don't restrain it, it will swivel sideways. But if you restrain it, you will feel yourself pushing against the exact same kind of pressure that you feel when you push two like magnetic poles together. In the latter case, it is not technically centrifugal force in the sense of being a radially outward force. It is a mutually orthogonal sister concept to centrifugal force involving the axial deflection of a transverse motion. It is not recognized in the textbooks. It is effectively an axial Coriolis force.

Then there is the transverse Coriolis force which is effectively a transverse centrifugal force with the circulation ω doubled, due to the rigidity of space in that particular mode of motion. The transverse Coriolis force is tied up with conservation of angular momentum. If you want to feel transverse Coriolis force, you will need some difficult to make contraption involving a turntable with a radial groove with some object in outward radial motion along the groove. When you turn the turntable, you will find that you are having to put in work in order to sustain a constant angular speed, due to the outward radial motion of the object along the groove. That will probably feel something similar to pushing two like magnetic poles together, or like force precessing a spinning gyroscope and restraining it to plane of forced precession.

This entire topic is about three mutually orthogonal pressures which arise in a solenoidal dipolar field. The radial one is being masked nowadays by a hall of mirrors. The transverse one is also being masked by a hall of mirrors and it has broken loose at its hinges and been allowed to swing around in the wind like a weather cock. The axial one is not even recognized in the literature, which is why they can't explain gyroscopes and rattlebacks. David Tombe (talk) 13:47, 24 May 2009 (UTC)

You're self-evidently a crackpot. Go away and take your ridiculous OR with you. This is not the place to posit new 'theories' of physics. We've much better things to do here.- (User) Wolfkeeper (Talk) 02:09, 25 May 2009 (UTC)

Wolfkeeper, you have been a major part of the problem on this page and I'm very sorry that the administrators can't see right through you. David Tombe (talk) 11:28, 25 May 2009 (UTC)

Look, centripetal force is just the second differential of the motion with respect to an inertial frame. You set x=r cos (wt) and y = r sin (wt) and differentiate twice. You end up with an acceleration along the radius vector; and there's a force from that F=ma. Reactive centrifugal force is just Newton's third law on that centripetal force. That's all there is to it. There's no expansion, nothing. It's trivial calculus. There's no mystery. And 'fixed stars' are nothing to do with it- any inertial frame works fine, even if the stars are whizzing past your ears provided you're moving in a straight line and gravity is not too strong. This is physics that any 16 year old that has done basic calculus can handle. I don't know where you've got it into your head that this is the cornerstone to physics, but really, it's just trivial. Unbelievably trivial.- (User) Wolfkeeper (Talk) 21:10, 25 May 2009 (UTC)

Mach's principle? Mach's principle is wrong. There has never been an experiment that backs it up, and General Relativity flatly denies it. Einstein liked it, but couldn't get it to work either. Mach's principle is a guess; that failed. There is no expansion.- (User) Wolfkeeper (Talk) 21:10, 25 May 2009 (UTC)

Wolfkeeper, the equation that we are arguing about is this one,

$m{\ddot {r}}-mr{\dot {\theta }}^{2}=f(r)$

It is the central force equation 3-12 from Goldstein. It contains both a centripetal force term $f(r)$ and a centrifugal force term $mr{\dot {\theta }}^{2}$. In the special case when the two are equal in magnitude, the $m{\ddot {r}}$ term will be zero and we will hence have circular motion. In the rotating frames of reference approach to centrifugal force which you seem to prefer, they are advocating that circular motion arises in conjunction with a net inward centripetal force. So something is seriously wrong. One of these two approaches must be wrong. As for Newton's reactive centrifugal force approach, it is definitely wrong because centrifugal force and centripetal force are not in general equal in magnitude. There are three approaches to this topic, and only one is universally correct. The other two approaches can be correct for limited applications. You are advocating that there are only two approaches to this topic, both different, but both equally correct. You can now see that the planetary orbital approach is a third way that conflicts with the rotating frames approach, and so the statement in the introduction which says that there are two approaches to this topic is wrong. And it has been put there deliberately as a corrupt bureacratic means of denying the existence of the third way, which is in fact the only universally correct way. David Tombe (talk) 22:11, 25 May 2009 (UTC)

All I see is a crank claiming that Newton didn't understand Newtonian mechanics, and the crank is claiming that their cranky views ought to be written into the wikipedia as if they were gospel fact, and they are busy trying to twist any and all references to try to support their incredibly suspect views.- (User) Wolfkeeper (Talk) 01:06, 26 May 2009 (UTC)

Wolfkeeper, There are sources cited on the main article which clearly expose the conflict between Newton and Leibniz. Leibniz succeeded in not only establishing the inverse square law for gravity, but also the inverse cube law for centrifugal force. Leibniz had an equation which put the two together and demonstrated that planetary orbits arise out of these two forces working together in tandem. Circular orbits only occur in the special case when these two forces are not equal and opposite. When the two are not equal in magnitude, then we get non-circular orbits. When Newton saw Leibniz's equation, he behaved similarly to yorself. He tried to sabotage it. He invoked the specious argument that centrifugal force is an equal and opposite reaction to centripetal force, and hence the specious 'reactive centrifugal force' concept was born. But the evidence is that Newton knew otherwise and that he was only trying to denigrate Leibniz's work. Newton knew fine well that centrifugal force is not always equal in magnitude to centripetal force. Newton only got as far as establishing the inverse square law relationship for gravity, and he was obviously intensely jealous of Leibniz for having beaten him to the full planetary orbital relationship. It's sad to see that you have been successfully fooled by Newton's 'reactive centrifugal force' concept, which was merely a jealous reaction to Leibniz's equation. David Tombe (talk) 08:13, 26 May 2009 (UTC)

Guilty as charged, I'm fooled by Newton's mechanics (actually I'm even more fooled by GR and QM). I think it's time for another ban for somebody who has been trying to get equal space for discredited theories and has been spamming talk pages over a considerable period. This is never, ever going to work. Nobody is ever going to add more than a passing mention in the history section for Leibniz anywhere in the wikipedia.- (User) Wolfkeeper (Talk) 12:52, 26 May 2009 (UTC)

Wolfkeeper, Isn't it funny then how Goldstein still uses Leibniz's equation to solve the planetary orbital problem? Here's Goldstein's equation 3-12,

$m{\ddot {r}}-l^{2}/mr^{3}=f(r)$

Now imagine that equation when the centripetal term f(r) is the inverse square law gravity force. Then compare it with Leibniz's equation [1] and tell me the difference. It's not history. You're quite wrong on that point. David Tombe (talk) 21:06, 26 May 2009 (UTC)

Never mind that trivial equation. Tell us again how Newton's third law doesn't apply in orbital mechanics problems; how the reactive centrifugal force and the centripetal force are different: "Newton knew fine well that centrifugal force is not always equal in magnitude to centripetal force."(sic).- (User) Wolfkeeper (Talk) 02:53, 27 May 2009 (UTC)

That's a bit of a red herring; nobody is claiming the reciprocal r cube term is the reactive force, right? David is referring to the fact that Newton vascillated between the different interpretations of centrifugal force, as did most others for another hundred years or so. The one interpretation, the reaction force, obviously has them equal; the other, which David is now referring to, is what Newton originally appeared to think, more like the Leibniz viewpoint, except that they didn't get the point that it only made sense in a co-rotating frame. The sources clearly indicate much confusion back then. Dicklyon (talk) 03:57, 27 May 2009 (UTC)

Thank you Dick, at least you are keeping the argument focused. As you correctly state, nobody here has been trying to say that Newton's 3rd law breaks down in planetary orbital theory. Wolfkeeper seems to think that the centrifugal force and the centripetal force are an action-reaction pair. They are not. Newton's third law holds in planetary orbits, but it holds over two bodies. Wolfkeeper doesn't understand this topic. He clearly can't grasp the second order differential equation in which the inward gravitational force and the outward centrifugal force both vary with different power laws, leading to stable conic section orbits. David Tombe (talk) 06:22, 27 May 2009 (UTC)

They are an action-reaction pair... by definition. Either you're defining it differently, fine, but we want that definition, right now, or you're denying Newton's third law, in which case you're a crank and you can STFU.- (User) Wolfkeeper (Talk) 17:26, 27 May 2009 (UTC)

Wolfkeeper, Newton's third law acts over two bodies. Centrifugal force and centripetal force are not in general equal in magnitude and they both act on the same body. Hence, on two counts, centrifugal force and centripetal force do not constitute an action-reaction pair. Nobody is denying Newton's third law of motion. You just don't know how to apply it properly. And by the way, this is one issue over which I have changed my position since this edit war began. Originally, I had not considered the concept of 'reactive centrifugal force'. When I did consider it, my first reaction was that it is an action-reaction pair. I was wrong however. At that time, your allies were quick to tell me that I was wrong. In fact I have the thread here. [2] See how the administrator put the onus on me to provide evidence that centrifugal force and centripetal force are an action-reaction pair. But now you are putting the onus on me to produce the evidence that it isn't. I have now changed my position on that matter. You are now at variance with your ally FyzixFighter. David Tombe (talk) 18:45, 27 May 2009 (UTC)

Coriolis Force and Conservation of Angular Momentum

This is largely for the benefit of SBHarris. It's a pity that we have to go to the bother of explaining something so basic that nobody else was even arguing about. But it's necessary in order to prevent any further proliferation of the confusion.

Kepler's second law of planetary motion is the law of constant areal velocity. It is equivalent to the law of conservation of angular momentum. When expressed mathematically, it takes on the form of a transverse equation in which the two terms sum to zero. The two terms are the Coriolis force and what has been referred to in wikipedia as the Euler force.

This means that in a non-circular planetary orbit or in any vortex, a radial motion will have a transverse deflection in conjunction with a change in the magnitude of the transverse speed, such that angular momentum is conserved. We have two opposite forces that mathematically cancel. One causes the radial direction to deflect into the transverse direction, while the other causes a change in the transverse speed. The former is the Coriolis force and the latter is usually called the Euler force in wikipedia articles.

SBHarris has been trying to tell us all that the Coriolis force has got nothing to do with the conservation of angular momentum. In actual fact, it has got everything to do with the conservation of angular momentum and it is important that SBharris is corrected on this point.

It would seem that SBharris's entire knowledge of this topic is based exclusively on the fictitious forces/rotating frames of reference approach. And that because he has no knowledge of inertial forces in polar coordinates or central force theory, he has decided that these alternative approaches to centrifugal force simply don't exist. David Tombe (talk) 22:11, 28 May 2009 (UTC)

Which should be fine, no? If they don't exist in any one reference frame, they can't be physically real, can they? See, that’s the problem when you get an equation and you decide what you want the terms to mean. That's boloney. In the case of a planet orbiting the Sun, from the inertial view, there’s only one force: gravity. That’s it. No Coriolis, no Euler, no centrifugal. Not two tangential forces that mathematically "cancel," except in your mind and when inhabiting some corner of some notation of your choosing. Boy, you really had to split things up to get away from the only real force in the problem, didn't you?

Sorry, but there's an easier way, and since it gives the same answers, it's also correct, and THREE of the forces you rely on are simple figments of your chosen notation. When the planet in a elliptical orbit draws near the Sun it will go faster—but that doesn’t mean there was or is some transverse or tangential “force” acting on it to make it go faster, like Martians or a rocket pushing on it in the direction of orbit. And likewise when it gets far away and slows down—that’s not due to an extra transverse force opposing it, in a direction to slow it down. Instead there are other good reasons for this behavior which involve no forces other than gravity. When the planet is nearest, it’s going too fast to stay in circular orbit so it draws away. As it does, it trades velocity for potential energy and slows. The “force” which causes this is merely the sun pulling in a direction which is not the same as the velocity vector—that’s it. No Euler, no Coriolus, no centrifugal. Just gravity.

Now, you can make things complicated. You posted this equation $2{\dot {r}}{\dot {\theta }}+r{\ddot {\theta }}=0$

or if you like: $2{\dot {r}}{\dot {\theta }}=-r{\ddot {\theta }}$

Now there are two terms here, and they are equal for any centrally acting force, and this a statement of conservation of angular momentum. But that does NOT mean (or prove) that both of these terms are forces. They’re just terms describing motion (do you see any F’s?). They only become terms which describe “forces” when you insist on making it so, by transferring to a rotating frame in which fictitious tangential forces appear. Or, by deliberately splitting these terms out of the force equation (which you’ve already done) and thus totally ignoring the tangential components of the single gravitational force which causes both of these accelerations. There’s a tangential component of G which causes $+{\ddot {\theta }}$ when the planet approaches perihelion, and another which causes $-{\ddot {\theta }}$ as it recedes toward apihelion. In neither case does Coriolis, Euler, or any kind of centrifugal component have any rule. There’s just centripetal gravity. Which, of course, if it is exactly the right strength, with the right mix of $G$, $M$, $r$ and ${\dot {\theta }}$ in your problem, will produce a circular orbit, with never any tangential force component even in your notation, and thus $2{\dot {r}}{\dot {\theta }}=-r{\ddot {\theta }}=0$ SBHarris 23:47, 28 May 2009 (UTC)

I don't understand the argument here. Both approaches exist, and give the same result. In the inertial frame, the only force is the centripetal gravity. In the co-rotating frame, you've got the fictitious forces, but you can ignore the Coriolis and Euler as they cancel, making it a 1D problem in r (by definition of co-rotating, but you can work out the details and see the cancellation if you like). Of course if you look at transverse speed you can connect it to these transverse forces if you like, and that's one way to think about conservation of angular momentum, but by no means necessary. Dicklyon (talk) 00:12, 29 May 2009 (UTC)

The argument is due to Mr. Tombe really believing these fictitous forces exist in some real way-- not just Euler and Coriolis, but centrifugal force also. And that without them, there would be no conservation of angular momentum. So when the skater pulls her arms in, and twirls faster, he really does NOT have an alternate way to explain this, except with these other forces. SBHarris 01:34, 29 May 2009 (UTC)

They are inertial forces that are built into the inertial path. In the transverse direction, relative to any arbitrarily chosen point in space, the two transverse inertial forces cancel mathematically, but they do not cancel physically. The can both be individually observed. One of them acts to change the speed, whereas the other acts to change the direction. SBHarris is in total denial of the reality of these inertial forces because he has allowed the topic to become confused to the extent that he thinks that we need to have a rotating frame of reference in order for these forces to exist.

Dick says that we can ignore the two transverse inertial forces because they cancel mathematically. But we cannot entirely ignore them. It's like saying that we can ignore two equal and opposite pressure forces that act on an object. They may cancel mathematically but they produce a pressure, or even a torque. Furthermore, the conservation of angular momentum that follows from the mathematical cancellation is carried into the radial equation. The radial equation is not a one dimensional equation. It is the same equation as the equivalent one dimensional problem, but it is describing a radial motion that is rotating with a fixed angular momentum.

I think the above red sentence says it all. The radial equation is rotating. So it is just another expression of the rotating frame transformation. −Woodstone (talk) 14:27, 29 May 2009 (UTC)

The Coriolis force is visible in all vortex phenomena. As something moves radially inwards, it gets defelected into the transverse direction. That is the Coriolis force. But SBHarris is focused on another version of the Coriolis force as appears in other textbooks. SBHarris's Coriolis force is actually the apparent deflection of a motion that is observed from a rotating frame of reference. But SBHarris's Coriolis force cannot explain why cyclones are visible from outer space. We need to have the real inertial Coriolis force to explain that. David Tombe (talk) 10:09, 29 May 2009 (UTC)

Woodstone, The maths in the polar coordinate equations is a more accurate version of the maths in the rotating frame transformation equations. It carefully segregates the terms into radial and transverse. Rotating frames of reference are a red herring in all of this. A rotating frame of reference is only significant when it is a physical reality, and when the inertial forces are opposing the dragging forces. The best example would be a rotating turntable in a state of angular acceleration with a marble rolling out along a radial groove. In that case, dragging forces would have to work against all three inertial forces. The three inertial forces are of course the centrifugal force, the Coriolis force, and the so-called Euler force. The inertial forces have got absolutely nothing to do with rotating frames of reference. They are built into the inertial path.

Gaspard-Gustave Coriolis was unto this with his category 1 supplementary forces. But he cocked up on the Coriolis force because he deduced its existence from the mathematical transformation equations and failed to link that term with the appropriate effect in category 1. The result was that we are left today with the Coriolis force in category 2, and swinging loose at the hinges like a weather cock.

The other branch of physics that deals with this topic correctly, ie. polar coordinates and central forces, is being suppressed from this article. Rotating frames of reference are an extra factor on top of the inertial forces and they are being handled wrongly. Coriolis force, which is a transverse inertial force, is being used to account for the apparent deflection which occurs when a motion is observed from a rotating frame of reference. That apparent deflection is not what Coriolis force is all about. Coriolis force is the real transverse force that is tied up with the conservation of angular momentum. David Tombe (talk) 15:39, 29 May 2009 (UTC)

Split in interpretation

Brews, the maths describes the inertial effect. But the textbooks are split on the interpretation of the maths. Some chapters accept that it is the inertial effect (eg. Goldstein, Shankar). Other chapters wrongly think that the maths is describing the apparent deflections that are observed from rotating frames of reference. The chapters that treat the maths as dealing with the inertial effect are being suppressed, principally by FyzixFighter and dicklyon. The chapters that treat the maths as dealing with the apparent deflection are in vogue. In truth, a rotating frame of reference does nothing more other than to superimpose a circular motion on top of the already existing inertial path, and its the inertial path that contains the inertial forces. FyzixFighter is playing a silly game with references and trying to make out that chapters that deal with central forces and polar coordinates really mean rotating frames of reference but without saying so. He is succeeding in pulling the wool over everybody's eyes. David Tombe (talk) 20:12, 29 May 2009 (UTC)

It seems to me that another way to say what you are saying is this:

There are two viewpoints. One is the viewpoint of the inertial observer watching an object in curved motion. The second is the viewpoint of the raotating observer, who sees the motion of the object differently, because their motion is admixed with the true motion of the body. You (D. Tombe) prefer to take the first viewpoint. But textbooks are split, some taking the second (rotating) viewpoint. Is that a correct entry point to this discussion? Brews ohare (talk) 03:21, 30 May 2009 (UTC)

No, that's not right. There's no centrifugal force in the inertial frame; it appears only in the co-rotating frame, that is, the frame where r-double-dot is acceleration. Not that I have any consistent interpretation of David's POV, but it's not that. Dicklyon (talk) 03:26, 30 May 2009 (UTC)

Brews, it's nearly right. One minor point that you have overlooked is that we don't need to have curved path motion for centrifugal force. Curved path motion is a product of centripetal force.

But yes, you are essentially correct. One part of the literature deals with the inertial forces in terms of polar coordinates and central force problems (such as the Kepler problem). No rotating frames of reference are involved.

Another part of the literature deals with the inertial forces in relation to rotating frames of reference. I have been arguing that rotating frames of reference have the effect of adding an additional circular motion on top of the inertial path, but that the inertial forces are actually in the inertial path anyway.

Hence, I have effectively been arguing that 'rotating frames of reference/fictitious force' as a topic, is rubbish. The superimposed circular motion is not an inertial force. It is an apparent deflection. I have further been arguing that the so-called rotating frames transformation equations, when properly scrutinized are in actual fact the very same maths as in the case of polar coordinates, and that they are not, as they claim to be, rotating frame transformation equations. I explained that last year. When you scrutinize it in depth, what has ostensibly been sold as a motion relative to the rotating frame and a motion of the rotating frame, is actually only the radial and the transverse components of the inertial forces.

But we are not arguing about that anymore. I have conceded that this faulty approach is prolific in the modern literature and I am not attempting to block its appearance as prime of place in the article. What I am trying to do is get a recognition of the 'inertial' approach in the main article. You and Wolfkeeper both recognize that this third way does exist and can be backed by references.

But FyzixFighter and dicklyon both have a vested interest in suppressing references to this third way. Dick keeps saying that the third way is simply the co-rotating frame of reference. That is his own opinion. But the sources that deal with the third way don't need to involve rotating frames of reference in the analysis. Woodstone seems to think that because a radial vector is rotating that this means that we are observing from a rotating frame of reference. That is the height of nonsense. Centrifugal force is induced by absolute rotation. It is not something which needs to be observed from a rotating frame of reference. Rotating frames are only significant if they are physically real and if there are constraining forces which entrain a motion. In that case, the inertial forces, which exist anyway, can be felt pushing against the constraining forces. David Tombe (talk) 11:41, 30 May 2009 (UTC)

Straight-line motion at constant speed

So let's focus on a frame that is not rotating, and an object. Let the object move at constant speed in a straight line. Its acceleration is zero, as neither speed nor direction vary with time. Newton's laws say the net force is zero. I think we all agree on this one. However, as I understand you, if I choose an origin from which to observe, the body has an angular motion from that origin, and therefore the net force from that origin is comprised of a balanced centripetal and centrifugal force with respect to that origin. Is this so?

Assuming this is the case, it follows that neither the centripetal nor the centrifugal forces are "fundamental" forces inasmuch as they vary with the choice of origin, and do not appear in a vector formulation at all, but only when a coordinate origin is introduced. Yes? Brews ohare (talk) 14:31, 30 May 2009 (UTC)

Brews, there is no centripetal force involved in the straight line motion. The straight line motion contains a centrifugal force, a Coriolis force, and an Euler force relative to any arbitrarily chosen points in space. The value and direction of these three forces is different for every origin. These are inertial forces. Their physical nature is subtley different from that of other forces. They are properties of space itself. The centrifugal force is radial to the chosen origin whereas the other two are transverse, equal and opposite in magnitude, and lead to the law of conservation of angular momentum. David Tombe (talk) 16:30, 30 May 2009 (UTC)

Mathematically, based simply upon the fact that the motion proceeds with neither change of speed nor of direction, in a vector formulation:

${\ddot {\boldsymbol {r}}}=0\ ;$

Only if we apply a centripetal force that is equal and opposite to the centrifugal force. David Tombe (talk) 16:35, 30 May 2009 (UTC)

David: This equation must be valid regardless of circumstance for straight-line motion. It is what a straight-line motion at constant speed means. Brews ohare (talk) 17:12, 30 May 2009 (UTC)

Not with respect to a position vector. You are thinking of displacement vectors. If your equation is true for that position vector, we need a centripetal force to balance the centrifugal force that is built into the inertial path. David Tombe (talk) 17:19, 30 May 2009 (UTC)

Are you suggesting I need an origin so the particle is located at r (t) − r_o? That's OK, for any fixed r_o I obtain the same result that :${\ddot {\boldsymbol {r}}}=0\ ;$. Brews ohare (talk) 17:55, 30 May 2009 (UTC)

Choosing an origin:

${\frac {d^{2}\mathbf {r} }{dt^{2}}}=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+{\frac {1}{r}}\quad {\dot {\overbrace {r^{2}{\dot {\theta }}} }}\quad {\hat {\boldsymbol {\theta }}}$

In the radial direction Newton's law is:

$({\ddot {r}}-r{\dot {\theta }}^{2})=0$

This is true, only if there is no centripetal force. David Tombe (talk) 16:36, 30 May 2009 (UTC)

David: This equation is just straight-line motion in polar coordinates; there is no room for discussion here. Brews ohare (talk) 17:12, 30 May 2009 (UTC)

Brews, those polar coordinate expressions have to be properly applied to a given physical scenario. That's what equation 3-11 is all about. It ties up those polar expressions with the central force problem. You cannot deduce any physical scenario from those expressions in isolation. David Tombe (talk) 17:22, 30 May 2009 (UTC)

David: You are suggesting that kinematics does not work? There is no need to introduce specific balancing forces to obtain straight-line motion; all we need to know is that the net force is zero. Brews ohare (talk) 17:55, 30 May 2009 (UTC)

We name ${\ddot {r}}$ as "centripetal" force and $r{\dot {\theta }}^{2}$ as "centrifugal" force. With these definitions, the statement that the centripetal and centrifugal forces balance in straight-line motion is irrefutable.

No Brews, this is where you are getting mixed up. For the general radial equation, see equation 3-11 in Goldstein. The centripetal term is denoted by f(r). The ${\ddot {r}}$ term is simply the expression for the second time derivative of the radial distance in general, irrespective of what forces are acting in particular. David Tombe (talk) 16:39, 30 May 2009 (UTC)

We are discussing straight-line motion here, not a central force problem. Brews ohare (talk) 17:12, 30 May 2009 (UTC)

In the transverse direction:

${\frac {d}{dt}}\left(r^{2}{\dot {\theta }}\right)=0$

How'm I doin'? Brews ohare (talk) 15:00, 30 May 2009 (UTC)

Correct. David Tombe (talk) 16:40, 30 May 2009 (UTC)

Next, I'd say there can be no objection to any of this, and we all would agree with it. However, the choice of terminology (that is, ${\ddot {r}}$ as "centripetal" force and $r{\dot {\theta }}^{2}$ as "centrifugal" force) is not universally adopted. Moreover, both of these "forces" are coordinate dependent (do not satisfy the conditions of Galilean invariance) and neither can be attributed to any source (they are not fundamental in the sense of the Standard Model, for example). So we can argue over choices of names for things, and that is what the argument is all about. Eh? Brews ohare (talk) 15:07, 30 May 2009 (UTC)

Brews, to a certain extent it's an argument over names. But anybody who has any comprehension of the topic will not be in any doubt that the $r{\dot {\theta }}^{2}$ is the one and only centrifugal force that is an outward radial force, induced by absolute transverse motion and which is the inverse cube law term in Leibniz's equation. David Tombe (talk) 16:49, 30 May 2009 (UTC)

There is no "one and only" when multiple definitions exist. Brews ohare (talk) 17:12, 30 May 2009 (UTC)

As a final observation, I'd say that the terminology (that is, ${\ddot {r}}$ as "centripetal" force and $r{\dot {\theta }}^{2}$ as "centrifugal" force) is used not only by David, but by an entire body of literature that approaches mechanics problems from a Lagrangian point of view, where these terms become "generalized forces", thereby releasing them from any requirement that they be either Galilean invariant or fundamental in the sense of the Standard Model. These authors are particularly evident in the field of robotics, where generalized coordinates are very useful to describe the configuration of machines with many articulated joints. The Lagrangian formulation is so familiar to these authors that the word "generalized" is very often omitted from "generalized forces" in their discussions. Brews ohare (talk) 15:17, 30 May 2009 (UTC)

Which is why we need to mention the third way in the introduction. Have you ever queried as to why FyzixFighter and dicklyon are so determined to suppress any mention of the third way? David Tombe (talk) 16:49, 30 May 2009 (UTC)

The work of Goldstein on the two-body problem also may be viewed as a formulation from a Lagrangian viewpoint (which is a rather common approach in advanced mechanics texts), and so the terminology (that is, ${\ddot {r}}$ as "centripetal" force and $r{\dot {\theta }}^{2}$ as "centrifugal" force) in Goldstein may be viewed as a formulation in terms of "generalized forces", and thereby released from any direct consideration of rotating coordinate frames. Nonetheless, the Lagrangian approach is mathematically equivalent (in this two-body problem) to choosing a co-rotating frame. Brews ohare (talk) 15:30, 30 May 2009 (UTC)

Yes and No, The concept of a rotating frame of reference totally confuses the topic, and it is not necessary. The inertial forces are absolute forces measured relative to the inertial frame. It's true that the centrifugal force only exists when the radial vector is rotating, but that has got nothing to do with rotating frames of reference. Gravity rotates too in a planetary orbit and nobody ever suggests that we need to be in a rotating frame to observe it.

A rotating frame simply adds an extra apparent transverse superimposition on top of the inertial path. David Tombe (talk) 16:49, 30 May 2009 (UTC)

David: You are not reading this carefully; it says that in the Lagrangian approach there is no need to discuss frames of reference because instead the "generalized coordinates" are introduced, which in turn lead to the "generalized forces". However, as with many such problems, the equations may be arrived at by different methods, and one of these is the method based on a co-rotating frame. It isn't necessary, it is just one possibility. Brews ohare (talk) 17:12, 30 May 2009 (UTC)

Brews, I'm not even sure that it is a possibility. I think it's a travesty. Rotating frames of reference serve only one purpose. They superimpose a circular motion on top of the already existing inertial path. The inertial forces are already built into the inertial path. It's not a question of not needing rotating frames in order to obtain the inertial forces. The rotating frames are nothing to do with the inertial forces. That Lagrangian method is correct in that respect because it pulls the inertial forces directly out of space without any mention of rotating frames.

Can't you see that 'rotating frames of reference' as a topic is rubbish? And when you realize that a topic in the curriculum is rubbish then you stop talking about it and you stop writing about it. You leave that for others to do.

David: No, I don't share that view. My view is that there are two contrasting perspectives: the viewpoint of Newton that involves "real" and "fictitious" forces, and the viewpoint of Lagrangian mechanics, which largely ignores the whole idea of "forces" and just deals with "generalized forces" that are double time derivatives of "generalized coordinates", and may not even have units in newtons. Brews ohare (talk) 18:17, 30 May 2009 (UTC)

At the moment, the big problem that we are facing is that a young generation have learned about centrifugal force within this nonsense context and they now want to parrot it on wikipedia. These people have never learned about the alternative approaches and so they will not entertain them. It's a kind of an attitude based on 'I didn't learn it in my education, so nobody will learn it'. That's what we're up against here. These people are absolutely fascinated with the extrapolation of the rotating frames centrifugal force to its most ludicrous conclusion, which is of course the 'radial Coriolis force'. They are fascinated with this nonsense and so they want to parrot it at the expense of having a serious article on centrifugal force such as that which occurs as the outward expansion force in planetary orbits. These people are part of the 'everything is relative/anything goes' generation and they are out to destroy this article. David Tombe (talk) 17:35, 30 May 2009 (UTC)

After all the discussion in this section I come to the conclusion that there is no discrepancy between David's formulation and the fictitious forces one. The latter are vector equations, the former are scalar equations on some curvilinear coordinate values. There may only be some differences based on what terms in these equations carry which names. Since the r coordinate used is co-rotating, the object's speed in the corresponding rotating reference frame only generates a tangential Coriolis component, and the radial component David objects to will not occur in the rotational frame. −Woodstone (talk) 22:36, 30 May 2009 (UTC)

Woodstone, yes the mathematics behind the two formulations is identical. That's what I've been trying to say all along. But in the polar coordinates/central force theory, it is done more rigorously such as to confirm the Coriolis force to the transverse direction. In the rotating frames/fictitious approach, the final result is left silent on the issue, even though anybody working through the derivation would have realized that the Coriolis term came from the radial component of the velocity, and hence the v itself is a radial v, and so the 2v×ω term must be in the transverse direction.

The latter begins ostensibly as a rotating frames transformation derivation. We begin with a polar position vector for a point A. We then root that vector through a fixed point, B, on an imaginary rotating frame of reference. This splits the vector into two components. The first component, OB, then becomes the position vector of that fixed point on the rotating frame, and the second component BA becomes the position of the point A, relative to the fixed point on the rotating frame. When we shrink the triangle to the infinitessimal limit for calculus purposes, the two components end up as the two mutually othogonal inertial forces, one radial and one transverse, and the imaginary rotating frame of reference becomes totally irrelevant.

So what we have is people studying fictitious forces and rotating frames of reference and thinking that they have got the transformation equations which deal with cannons firing missiles from rotating turntables, or for the cyclones in the atmosphere relative to a pole centred coordinate frame. In actual fact, all they have got is the equation for the inertial forces in the inertial frame of reference.

It's a monumental cock up in modern applied maths. I remember when I was first shown that derivation in 1980 and shown that the Coriolis force was free to swing around in any direction. I was instinctively suspicious at the time, although I only investigated it properly a few years back. David Tombe (talk) 23:02, 30 May 2009 (UTC)

Straight line motion with accelerating force

It seems to me that we can reduce this argument to even more fundental terms. An object moving inertially will move in a straight line, absent any forces. Or, if you wish to fix your inertial coordinate system upon it, will be motionless. Now, let us introduce a single external force F. In the inertial system the object takes on an acceleration a = F/m. In the free body diagram of the body we draw just one force arrow, with no opposing "inertial force." But this is already a bit wonky, as where do we account for the opposing force of inertia, in this inertial frame? So far as I can see, in the frame where the body accelerates, IT ISN'T THERE. Or rather, is there but isn't acknowledged. So maybe you physics wonks can tell me where it went?

Okay, now transfer to the accelerated frame with the accerating object, so that the object with the force applied is at rest. It could be a box in the cargo hold of an accelerating spaceship, and we could be on the spaceship with it. Now, in order to explain the fact that it is at rest in our accelerated spaceship frame, we need two opposite and opposing forces on it. One is the same force we had before, now provided by the spaceship floor. The other is now a "fictitous" intertial force which we imagine is trying to push the object against the floor, as though it had weight. Something that looks like artificial gravity. Now, here's what bothers me: that inertial force is always there, opposing the accelerating force. Even in the inertial frame where the box accelerates with the rocket, we see that the rocket pushes the box, AND THE BOX PUSHES BACK reactively on the rocket floor. Yet the reactive back-pushing inertial force we refuse to acknowledge in the inertial frame. We only "see it" (notice it??) in the accelerated frame, where we suddenly "need" it to "explain" why the box doesn't move (now our free-body diagram has two oppositely facing arrows-- one for the floor and one for the box's artificial weight). It seems to me that this neglect of a reactive force which we DO actually see in the inertial frame (this "inertial force" which resists accelerating forces), is somehow not right. This inertial force we ignore in the inertial frame BECOMES the "fictitous" "artificial weight" force, which we no longer ignore in the accelerated frame. Say what? Not fair. This force doesn't somehow "appear" in the accelerated frame-- it was always there. You see it as a reactive force even in the inertial frame, but ignore it. I think this is somehow related to what Tombe is saying. Except I've removed all components of off axis forces from it, so it's clear we're no longer going in circles. So to speak. And angular momentum and all that are gone. We're just talking about linear motion. SBHarris 19:33, 30 May 2009 (UTC)

You wrote; An object moving inertially will move in a straight line, absent any forces. Now, let us introduce a single external force F. In the free body diagram of the body we draw just one force arrow, with no opposing "inertial force."

The important thing here is the phrase "in the free body diagram". There is an equal and opposite reaction somewhere but it doesn't get included in the free body diagram by definition. There is no contradiction here. The purpose of the Free body diagram is to isolate one object to solve for force, acceleration, etc. If the force is due to a gravitational interaction with another object then there is an equal force exerted on that other object. It's hard to imagine but every object near Earth is exerting a gravitational force on Earth that is equal and opposite to what Earth's gravity exerts on it.-Crunchy Numbers (talk) 20:09, 30 May 2009 (UTC)

You wrote; (now our free-body diagram has two oppositely facing arrows-- one for the floor and one for the box's artificial weight)

If you draw a free body diagram of a box resting on the floor of a linearly accelerating spaceship there is only one force acting on the box: The force of the floor that accelerates it. The opposite pair to this is the force the box exerts on the floor due to its inertia but this is part of the fbd of the ship since this force is acting on the ship. That is why a spaceship that is loaded with heavy boxes will require more force from its engine to accelerate. It has to accelerate all those boxes.-Crunchy Numbers (talk) 20:28, 30 May 2009 (UTC)

No. If you draw a free body diagram in the frame of the ship for the box, there are now two (half) arrows on the box pointing in opposite directions, since the box is not accelerating, thus net force on it must be ZERO. The fact that the weight arrow also has another end which points at the ship, is merely Newton's third law-- real force arrows have two equal ends. But this two-ended "weight-weight reaction" arrow between box and floor exists whether you look at the box in inertial frame OR ship frame. It's the same in either frame. What's missing is the new inertial force which exists only in the ship frame, which is an arrow which doesn't HAVE another end, as it doesn't follow the 3rd law. So I think I may have answered my own question. A force arrow without two ends. Very odd! But all these "fictitous forces" are of this nature. SBHarris 21:56, 30 May 2009 (UTC)

Not so odd, because you forgot to mention how the engine produces force. If it's a simple rocket engine it will have to push back an exhaust flow of acelerated fuel particles. −Woodstone (talk) 22:24, 30 May 2009 (UTC)

No, this action-reaction pair happens with any "real" force, simply due to the nature of force and the conservation of linear momentum. Any force acts between two objects, for the same amount of time, and force*time on one end equals the same at the other. You could produce the force on the box floating in space by shining laser on it. Here the force is simple momentum transfer from reflected photon to box. However, the fictitous inertial force seen in the boxes accelerating frame, does not conserve momentum. There is no force between two objects, in the case of THAT force. As noted, it's a force-arrow with only one head. SBHarris 00:15, 31 May 2009 (UTC)

Coriolis on the Coriolis Force

I think that the problem with the Coriolis force began with Coriolis himself. There is a very interesting web link that is accessible in the history section on the main article. It goes into detail on Coriolis's original 1835 paper. Coriolis was of course interested in real forces that arise in water wheels. The article implies that he wasn't really into analysing the maths in depth, and that is why he probably made the mistake that has propagated into today's literature. Coriolis talked about 'supplementary forces' that act in rotating frames of reference. He divides these into category 1 and category 2 supplementary forces. The category 1 supplementary forces are clearly real physical effects. They are the inertial effects which work against the dragging forces that are needed to drag a motion with a rotating frame. These of course would correspond to all three inertial forces ie. the centrifugal force, the Coriolis force, and the Euler force. But it appears that Coriolis was only looking at the centrifugal force in this regard. There is no evidence that he was considering constrained radial motion on a rotating turntable. So basically, we get centrifugal force as his category 1 'supplementary force'. The problem arises when we go to Coriolis's category 2 supplementary forces. He deduces this purely from looking at the mathematical transformation equation. And the author believes that he didn't examine it very carefully. Coriolis sees the term that looks like the centrifugal force but which is multiplied by the factor of 2. He decides to call it the compound centrifugal force (which later became named the Coriolis force in his honour). But just like SBHarris, Gaspard-Gustave Coriolis allows this term to swing around freely like a weather cock. And this error still exists in the modern textbooks when the topic of 'rotating frames of reference' is being dealt with. On the other hand, chapters that deal with central force theory and polar coordinates have got the Coriolis force firmly fixed in the transverse direction as a real inertial force that is linked to the conservation of angular momentum. David Tombe (talk) 22:30, 28 May 2009 (UTC)

I'm just glad we have David Tombe to point out the entirely obvious error that every single mathematician and physicist since his day has missed. The world was waiting for David Tombe, we're so fortunate.- (User) Wolfkeeper (Talk) 23:58, 28 May 2009 (UTC)

If all the sources are in error, we have to report them as if they're correct. Those are the rules. But we can keep David's wisdom in our back pockets, and maybe use it when we write our own books, to set the world straight; or use it the next time nature calls. Dicklyon (talk) 00:00, 29 May 2009 (UTC)

Dick, I wasn't saying that we should be writing up against the sources. This whole topic is a mess in the literature, both past and present. We have to strike a balance between getting the facts right and lining them up with references. Coriolis's error is too great to amend in wikipedia, so we will just have to forget about it. Rotating frames of reference, despite all its faults, is the popular show of the day in the modern textbooks and so it has to go in, prime of place. But that doesn't mean that we have to suppress the many alternative references which treat centrifugal force and Coriolis force as inertial forces in polar coordinates without mentioning rotating frames of reference. David Tombe (talk) 10:15, 29 May 2009 (UTC)

Wolfkeeper. Your bias is showing again. Perhaps you are the problem and should recuse yourself form editing this article. You obviously are biased and have an unfair attitude. I suggest you not participate in these didcussions anymore.72.84.66.18 (talk) 14:01, 29 May 2009 (UTC)

Yeah, and how are they suppose to behave? They've been trying to explain this same (simple!) stuff for months (years?) now, with absolutely zero progress (except maybe, usually improved civility from both sides). Mr Tombe is convinced that centrifugal, Coriolis and Euler forces are real (just have a look at http://www.wbabin.net/comments/tombe.htm), and will probably never change his mind... Well, it's a well-known (and sad) fact that you can keep discussing in vain forever with crackpots.--129.194.8.73 (talk) 14:41, 29 May 2009 (UTC)

The "reality" of fictitious forces is not at issue, I think. They are real as seen by observers in rotating frames, that is, real parts of the F=ma relation in such frames. It's lame to interpret "fictitious" as "nonexistent" or "unreal", instead of just as what it means. Dicklyon (talk) 14:57, 29 May 2009 (UTC)

Agreed my formulation was bad. What I meant was that you can always make them disappear by choosing an inertial reference frame and describing the system in a coordinate-independent way. Does Mr Tombe agree with the latter statement? If not, then I don't see what's the point of this unending discussion...--129.194.8.73 (talk) 15:42, 29 May 2009 (UTC)

The obvious conclusion in reading this article is that the editors have done a rather poor job of it. Blaming Mr Tombe is not going to solve that problem. If the editors dont understand what they are doing, getting rid of one of them isn't going to help. By the way. I have learned that when a person is called a crackpot, it usually means there is no effective answer to his arguments. So I think you are saying he is correct, since your only recourse is empty insulting talk. There does seem to be some argreement that when there is a real rotation the results are real forces, and when there is not a real rotation the results are not real forces. This seems to be the point of disagreement, how to explain this. There is a resistance to the idea of an absolute rotation, because such a concept has no mathematical reality. This is because physical space is thought to be mathematical space which is simply an abstract relation. The rub seems to be that this idea doesnt really work in the real world of physics. What Mr Tombe seems to be saying, and for which he is being insulted, is that physics deals in the real world and not mathematical abstractions. If the real world contradicts the math, then that is the physics of the problem. Mr Tombe's opponents seem to be asserting that the mathematical abstraction is the primary reality and that physical reality must give way to the mathematical interpretation of space even if it means a loss of physical comprehension. I dont think you will get this right until you realise that physical reality isn't what mathematicans and textbook writers want it to be so that the math looks nice and orderly in textbooks. One way to resolve this problem would be to not call this a topic in physics and make it simply a mathematical discussion that claims to be applicable to physics. That is really what modern textbook writers are doing. They are simply inventing a mathematical view of the world and they dont bother to ask if this interpretation fits the physical reality at all.72.84.66.18 (talk) 15:35, 29 May 2009 (UTC)

72.84.66.18 I've just explained to Woodstone in the section above how rotating frames of reference only have any meaning if they link to physical reality, such as in the case of a turntable. From reading about Coriolis's original paper, I'm sure that Coriolis was only concerned with real rotating systems and the dragging forces that are needed to overcome the real inertial forces. Unfortunately it seems that even Coriolis cocked up as a result of his superficial reading of the transformation equations. He didn't put 2 and 2 together, otherwise the Coriolis force would have been in his category 1 supplementary forces. The abstract concept of rotating frames of reference, without any real dragging forces is totally meaningless, and the transformation equations certainly don't apply to it. So in the modern textbooks, we have two topics that cover centrifugal force. One is correct and the other is nonsense. The correct one is being suppressed from this article, and the nonsense one is being promoted. Unfortunately, the nonsense approach dominates the modern textbooks. David Tombe (talk) 15:50, 29 May 2009 (UTC)

The underlying difficulty here is the connection between the math and the physics. The textbooks have the math and the physics in sync, and both agree and agree with observation. David has a gut feeling that cannot be applied except by him, and contradicts the usual mathematical methodology. David may get the right answers in actual cases (I'm not sure about that) but so does the textbook theory, and it has the advantage of being intelligible to (almost) anyone.

Attempts to get David to approach the math and point out how he disagrees never succeed. He insists upon the "Leibniz equation" and will not broaden the discussion to include (for example) a vector formulation of the issues. Brews ohare (talk) 18:33, 29 May 2009 (UTC)

Mr brews, If you guys are so in synch, then you should be able to come up with a decent article. You havnt and blaming Mr Tombe is not going to make the article less confusing. Before Mr Tombe intervened the article was incomprehensible nonsense. That is gradually invroving. So why not stop your silliness. You guys are simply not going to get it right until you accomodate Mr Tombe.72.64.49.78 (talk) 20:59, 31 May 2009 (UTC)

The Inertial Path

Brews, I think that we do need to get one thing quite clear. And that is that,

${\ddot {\boldsymbol {r}}}=0\ ;$

is not the inertial path. This equation solves to a circular motion with constant angular speed. The inertial path equation contains only the three inertial forces and with the proviso that the two transverse inertial forces are equal and opposite mathematically. Hence, the inertial path embodies conservation of angular momentum and also a net outward centrifugal force relative to the chosen point of origin. Your equation above has got that centrifugal force equally and oppositely cancelled by an applied centripetal force.

The inertial forces will be felt if we oppose them. Rotating frames of reference have got nothing to do with the topic. Gaspard-Gustave Coriolis was looking at physically real rotating frames of reference in which the dragging forces are opposed by the inertial forces. But he cocked up when he tried to put a physical interpretation on the maths. Hence the Coriolis force swings loose on the pole like a weather cock.

SBharris has been puzzling over the concept of inertia in linear situations. Once again, we don't need to tangle the problem up with frames of reference. It's the same situation physically. When we accelerate a body we encounter an inertial resistance. It is all built into Newton's third law of motion. SBharris points out that there is no recognition of this inertial resistance as a distinct force in linear situations.

The difference is that in linear situations we use cartesian coordinates and displacement vectors which mask the inertial forces. The inertial forces are built into Euclidean geometry. They only show up mathematically when we use polar coordinates. Some might argue that real forces are not dependent on which coordinate system we use. But these particular real forces are catered for under the law of inertia when we use Cartesian coordinates. That's why they are called inertial forces. David Tombe (talk) 22:43, 30 May 2009 (UTC)

The inertial path equation is the vector equation ${\ddot {\vec {\boldsymbol {r}}}}=0$, with trivial solution ${\dot {\vec {\boldsymbol {r}}}}={\mbox{constant}}$.

The scalar equation in coordinate r reading ${\ddot {r}}=0$ solves to a circle.

−Woodstone (talk) 23:01, 30 May 2009 (UTC)

Woodstone, you got the scalar equation right. But if we then consider the total vector equation with the two transverse components which mutually cancel, the vector equation ${\ddot {\boldsymbol {r}}}=0\$ will still be a circle of constant angular speed. In a circle of constant angular speed , both of the two scalar equations sum to zero. The inertial path does not have a centripetal force (unless we admit gravity as a contributor to the inertial path, as some people do). The inertial path is a straight line motion with a net centrifugal force. David Tombe (talk) 23:06, 30 May 2009 (UTC)

The vector equation falls apart in components as ${\ddot {r_{x}}}=0$ and ${\ddot {r_{y}}}=0$; each of them integrates the same way. Integrating once gives solution ${\dot {\vec {\boldsymbol {r}}}}={\vec {v_{0}}}$, with ${\vec {v_{0}}}$ a constant vector. Integrating again gives ${\vec {\boldsymbol {r}}}={\vec {v_{0}}}t+{\vec {r_{0}}}$ with ${\vec {r_{0}}}$ the initial position. Just as expected: straight line movement with constant speed. −Woodstone (talk) 08:47, 31 May 2009 (UTC)

Woodstone, I think that we are going to have to start again and specify from the outside whether we were talking about a polar coordinate equation or a cartesian equation. I had assumed that we were in polar coordinates, but now you have switched to cartesian coordinates. In polar coordinates, ${\ddot {\boldsymbol {r}}}=0\$ is a circular motion of constant angular speed. David Tombe (talk) 09:28, 31 May 2009 (UTC)

As seen above and also in polar coordinates:

${\frac {d^{2}{\vec {\mathbf {r} }}}{dt^{2}}}=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+{\frac {1}{r}}{\dot {\overbrace {r^{2}{\dot {\theta }}} }}\quad {\hat {\boldsymbol {\theta }}}$

So the vector equation ${\ddot {\vec {\boldsymbol {r}}}}=0$ falls apart into

${\ddot {r}}-r{\dot {\theta }}^{2}=0$    and    ${\frac {1}{r}}{\dot {\overbrace {r^{2}{\dot {\theta }}} }}\quad =0$, which solves to a straight line.

The vector equations solve to a straight line in both cartesian and polar coordinates. Only the scalar equation on a coordinate (not a vector) ${\ddot {r}}=0$ solves to circle. −Woodstone (talk) 09:48, 31 May 2009 (UTC)

Woodstone, yes you are correct. The radial component of the vector equation, ${\ddot {r}}=0$ contains the centrifugal term in isolation without any centripetal terms, and so we get straight line motion, which is the inertial path. In order to make it into circular motion, we need to add an equal and opposite centripetal force. When we add a centripetal term, whether or not equal and opposite to the centrifugal term, then we will have Goldstein's equation 3-11. By making this error, I was contradicting my own position. As you guessed, I was too much focused on the scalar equation. Thanks for pointing that out. David Tombe (talk) 12:35, 31 May 2009 (UTC)

David, just ignore what I said above. I had temporarily forgotten that in free-body diagrams we must "cut" all Newtonian double-headed force vectors at the middle, and only use one "end" of them on our free-body. Of course a Newtonian (follows third law) force will have another end, acting on some other object. But it's no concern of ours! The "fictious force" vectors that appear in accelerated frames are not third-law two-headed vectors. They only have one head! They don't follow the third law! They just appear, in order to give us a "reason" why accelerated objects seem to have "weight" in the accelerated frame. This is not due to the usual kind of force that operates between two entities. That's why I think it's fair to continue to regard these forces as "fictitious" or at least of another nature than the forces that follow Newton's 3rd law (the four classical forces of nature-- gravitational, EM, weak, and strong). These all have two ends. Fictious forces have only one end.

At any rate, with your system in which you're following an object traveling in a staight line in polar coordinates, if you are following it with an r and and it moves by you, the r will rotate. Thus ${\dot {\theta }}=nonzero$. So whenever this is non-zero you're now in a rotating system of coordinates, and will see those funny fictious forces with only one head, which don't follow Newton's third law. I recommend you rid yourself of them by translating to a system where ${\dot {\theta }}=0$ and then you'll find all objects in inertial paths that are linear, will probably have no forces acting on them at all. If they do, they'll be two Newtonian forces (two heads each) in balance. SBHarris 00:41, 31 May 2009 (UTC)

SBharris, we know that the inertial forces don't show up in Cartesian coordinates when we work in displacement vectors. But the law of inertia still holds. The physical nature of the inertial forces is beyond the scope of this article. Don't rule out their reality just because they don't show up in Cartesian coordinates. Inertial effects are well quantified by position vectors in polar coordinates.

Your problem is that you are focused on illusory scenarios that we observe in accelerated or rotating reference frames. Those apparent deflections and accelerations are not inertial forces, contrary to what it teaches in some chapters of textbooks. They are just apparent effects. The rotating frame transformation equations yield the inertial forces. They do not yield the apparent effects contrary to what they claim to be yielding. Hence we find the likes of Woodstone coming along and saying that the two approaches are the same for the co-rotating situation. But if we are dealing with co-rotation then we are not dealing with the apparent deflections that the topic is supposed to be all about. In co-rotation we are in effect dealing with the inertial forces that act on the object itself, and the rotating frame of reference becomes irrelevant. Polar coordinates is an alternative way to treat this topic, and there are references in the literature for it. I gave a good one in my edit to the introduction which FyzixFighter keeps removing. In my opinion, that is the only correct approach to this topic. But I am not trying to erase the other approach. On the contrary, FyzixFighter is trying to erase all mention of the polar coordinates approach. David Tombe (talk) 09:26, 31 May 2009 (UTC)

What the devil do you mean by "inertial forces"? If you mean these one-headed arrow "fictitious forces" which don't follow Newton's third law and are not caused by gravity, EM, strong or weak interactions (the basic forces of nature) then say so. SBHarris 17:48, 31 May 2009 (UTC)

Steve, the inertial forces are the three forces which arise in all absolute motion, including straight line motion. They are quantified when Newton's laws are written in polar coordinates. We do not need a rotating frame of reference in order to observe them. To answer your question, in my opinion, they are electromagnetic pressures, but that is entirely original research on my part. Nevertheless, I got my ideas from Maxwell's 1861 paper, which you stated last year in so many words to be rubbish. I can link the centrifugal force with the vXB force in electromagnetism. You can have a look at my article at [3]. But for wikipedia purposes all we need to know are the basic facts. And the basic facts are that centrifugal force is an inertial force which is built into all motion in the inertial frame of reference, and that there is a different centrifugal force for every chosen point of origin.[4] The textbooks don't claim to know the physical nature of the inertial forces.

Rotating frames of reference are all about apparent deflections. Those deflections, contrary to what the textbooks teach, are not accounted for by the inertial forces. The rotating frame transformation equations in actual fact yield the inertial forces, exactly as in the case of polar coordinates. The two sets of maths are identical. But the polar coordinate approach is more accurate and it doesn't get all tangled up with rotating frames of reference. The rotating frames approach is a dog's dinner which claims to be accounting for the wrong effects.

Some textbooks deal with centrifugal force the correct way. But unfortunately most of the textbooks that are up front on the science library shelves today are teaching it the rubbish way. Some editors here, who have only learned it the rubbish way, are insisting on suppressing all references to the alternative way. It's a case of 'I didn't learn it that way, so nobody will'. David Tombe (talk) 18:20, 31 May 2009 (UTC)

As I've tried to tell you, if d(theta)/dt ≠ 0 in polar coordinates, then you DO have a rotating frame and thus an accelerated frame of reference. Thus, the "three inertial forces" you talk about in polar coordinates are merely the 3 fictitious forces that arise from an acclerated reference frame, which is what you get whenever d(theta)/dt ≠ 0, which is all of your examples. And I'm beginning to think these "inertial forces" are something you just made up out of your head with no connection with any accepted concept in physics texts, except possibly for the fictitious forces which arise in accelerated rotating frames. "Law of inertia" is just your term for your own private physics. SBHarris 18:50, 31 May 2009 (UTC)

Steve, I didn't make up the law of inertia. And I simply can't see how you can possibly equate the absolute rotation of something with a rotating frame of reference. If I move in the inertial frame of reference, I will have an angular speed, in relation to a chosen point of origin, relative to the inertial frame of reference. I will hence have an outward radial centrifugal force relative to that point. I can't see where rotating frames of reference enter into the picture. David Tombe (talk) 18:57, 31 May 2009 (UTC)

Okay, but you have to make sure you have no rotating frame of reference. So the rule for that in polar coordinates (as in rectilinear also) is that the coordinate axes must be pointed at the fixed stars, in such a way that they (the stars) don't move. Straight line motion in such a system, if does not pass through the coordinate center, gives you many funny terms. However, they don't necessarily correspond with real forces, and in generally they do not if the straight line motion is of constant velocity. If you arrange your coordinate system so that motion passes through the coordinate origin (or you affix the origin to the object in motion at least at one point in its path) then "real" forces show up straightforwardly as terms where d(theta)/dt ≠ 0 for the r vector pointing from origin to object, and also in terms where d^2(r)/dt^2 ≠ 0. SBHarris 19:54, 31 May 2009 (UTC)

Steve, you mentioned a rule for deciding whether or not we have a rotating frame of reference. Who made up that rule? I have never heard it before. The fact that something is rotating does not mean that we are observing it from a rotating frame of reference. When something moves in the inertial frame, it will have an angular speed relative to any arbitrily chosen point, unless that point is along the line of motion. Rotating frames of reference don't enter into the picture. You will have to learn to cast aside this nonsense about rotating frames of reference.

In polar coordinates we then observe three inertial forces relative to that point. The issue of whether or not you consider these inertial forces to be real is irrelevant. The only thing that matters here is to describe them. I personally believe that they are real, but that is entirely beside the point.

Whatever the physical nature of these inertial forces is, they are relative in the sense that they are different for every chosen point of origin, but they are absolute in that their magnitude, for a chosen point of origin, is dependent on their absolute angular speed, relative to the background stars. That is the nature of them. For alot of people, it is a very puzzling nature because on the one hand it is merely a product of the inertial path, but on the other hand it can cause a real outward pressure. Wikipedia is not however the place to dig deeper and investigate this mysterious phenomenon. But unfortunately, the mysterious properties of centrifugal force confuse people to the extent that they can't accept the simple facts that are before their very eyes. David Tombe (talk) 21:15, 31 May 2009 (UTC)

You are kidding about never having heard the rule about how to tell if you're in a rotating frame or not, hey? It's part of making sure you're in an inertial frame to make sure you're not in a rotating frame. SBHarris 08:22, 1 June 2009 (UTC)

Steve, the inertial terms in the polar coordinates are relative to the inertial frame. Rotating frames of reference don't enter into it. If something is rotating, it doesn't mean that we are observing it from a rotating frame of reference. Once again I have to conclude that you are taking all your ideas from a topic in the textbooks which I consider to be totally flawed. You have totally closed your mind to my exposition of the flaws, and you have also totally closed your mind to other parts of the literature which deal with this topic correctly. This entire edit war has been because of people like yourself who, on being introduced to the topic of polar coordinates for the first time, simply sweep it aside as if it didn't exist. It's a case of 'I didn't learn this at school, so nobody is allowed to learn it'. You need to open your mind to the fact that this topic is dealt with in the literature in at least three contradictory manners. You have read one of these, and you have decided to buy it without any further investigation. You have bought the popular approach of the day, which I can assure you is wrong. David Tombe (talk) 10:45, 1 June 2009 (UTC)

The Reason For This Controversy

The reason for this controversy is, as Mr Tombe contends, scientific political correctness. The following citation clears this up. "The theory of relativity...has done much...but when it comes to centrifugal force, then, as Eddington himself says, the theory stops explaining phenomena and begins explaining them away." From The Skeptical Physicist", By Paul R. Heyl, The Scientific Monthly, 1938. This is exactly what is happening here. The attempt is to explain away centrifugal force to support the modern theory of relativity. The attempts to suppress objections to this explaining away of centrifugal force as are being done here are just an example of scientific political correctness. Mr Tombe is not a crackpot, he is not incorrect in his views, and the efforts of wikipedia to suppress his edits are offensive to an attitude of free scientific enquiry. The editors need to realise that there is a legitimate reason to object to the way this article has been written. The personal attacks and continued harrassment by wikipedia editors is rude and irresponsiblebehavior and demonstrates the unfair bias of the wikipedia system of editing.72.64.53.168 (talk) 11:57, 1 June 2009 (UTC)

That's basically true. But I am not even trying to suppress references to the approach which I disagree with. I am trying to get a proper recognition of centrifugal force as a concept which arises in polar coordinates and planetary orbital theory, which exists in numerous references according to the anonymous 63 -- -- -- --. I don't know what all these references are but I have no doubt that he knows what he is talking about, and besides, I do know some such references. From what I can see Brews ohare, Wolfkeeper, anonymous 63 -- -- -- --, and anynomous 72 -- -- -- --, as well as myself have all acknowledged the polar coordinates approach yet FyzixFighter, dicklyon, and Gandalf61 have been trying to write the introduction so as state that there are only two approaches to centrifugal force, as opposed to three. They should not be allowed to get away with this. Administrator intervention is needed. We cannot entertain specious arguments of the kind such that the references to 'centrifugal terms' are not references to 'centrifugal force'. This is cheap word play and it should be seen instantly for what it is. This issue should never have been allowed to get to the stage where certain editors can revert sourced edits freely and come out with such specious arguments. David Tombe (talk) 18:41, 1 June 2009 (UTC)

Two Meanings, or Three? (Curved Coordinate Systems)

The article currently states that the term centrifugal force can refer to two distinct things, either the fictitious force in a rotating coordinate system or the reactive force, but I think there's a third meaning that can be found in quite a few reputable sources. The third meaning is a fictitious force in stationary but curved coordinates. Quite some time ago this was discussed here, and numerous references from reputable sources were provided for this definition, but all this seems to have been expunged from the article. I think, for accuracy and completeness, this third definition should be restored.63.24.42.134 (talk) 00:26, 31 May 2009 (UTC)

63.24.42.134 I think that that's exactly what I have been trying to do. Only I would prefer to call it an inertial force and not a fictitious force. At the moment, there are two editors who are determined to restrict the reader to believing that only two approaches exist to centrifugal force. Another editor who has not been involved in the debate ie. Gandalf61 has backed them up to give them numerical superiority. I have supplied an unambiguous reference to Shankar but it is simply brushed aside. I have even informed Jimbo Wales of this blatant disregard of a reliable source, but so far I have not had any response. It seems that the administrators here have been fooled by the specious arguments that are being put forward by editor FyzixFighter, such as that a centrifugal term doesn't necessarily mean a centrifugal force. David Tombe (talk) 09:12, 31 May 2009 (UTC)

Here are two reputable references for the "third" meaning mentioned above:

"An Introduction to the Coriolis Force" By Henry M. Stommel, Dennis W. Moore, 1989 Columbia University Press. "In this chapter we have faced the fact that there is something of a crisis in intuition that arises from the introduction of the polar coordinate system, even in a non-rotating system or reference frame. When we first use rectilinear coordinates to understand the dynamics of a particle, we commit our minds to the simple expressions x" = F_x, y" = F_y. We think of the accelerations as time rate-of change [per unit mass] of the linear momentum X' and y'. Then we express the same situation in polar coordinates that partly restore the wanted form. In the case of the radial component of the acceleration we move the r(theta')^2 term to the right hand side and call it a "centrifugal force."

"Statistical Mechanics" By Donald Allan McQuarrie, 2000, University Science Books. "Since the force here is radial, it is convenient to use polar coordinates. Taking x = r cos(theta) and y = r sin(theta) [i.e., stationary polar coordinates] then... If we interpret the term [r(theta')^2] as a force, this is the well-known centrifugal force...

Many more references can be cited if necessary, but these two should suffice to prove that this "third meaning" is indeed present in the contemporary literature, so I don't see any legitimate reason to suppress it in the article.130.76.32.15 (talk) 19:31, 1 June 2009 (UTC)

I still fail to see how these support a "third meaning" independent and distinct of the other two. For example, from the Stommel reference, on the previous page he states:

Sometimes equation 2.14a is written with one of the acceleration terms on the righthand side. The term r(phi-dot)^2 then looks like a force, and it actually has a name: "the centrifugal force" [per unit mass]. It is always positive and directed away from the origin. But it is really not a force at all, and so if we want to make use of it in a formal sense, then we could call it a virtual, fake, adventitious force.

Note that at the beginning of section 1.2 Apparent Force he clearly ties "adventitious forces" to observations in a rotating frame:

We now want to compare the visual image of what we have just seen in figure 1.1 with what we observe when we are moving with a uniformly rotating reference frame, when there is in fact no force turned on. The observer intuits that a force is acting even when there is no force. The impression is so strong that it is actually useful, and leads to the invention of the class of nonexistent, apparent, virtual, fake or adventitious forces call "centrifugal force" and "Coriolis force."

The connection is much clearer when he does the full apparent force derivation in ch 3, where the transposition of the r(phi-dot)^2 term to the righthand side corresponds to transforming to a rotating frame where Sigma=phi-dot, ie a corotating frame.

"Newtonian mechanics" by AP French (1970) describes the motion of a tether ball from two frames, a stationary frame S, and arotating frame S' that rotates with the same rotational speed as the ball:

1. From the standpoint of the stationary (inertial) frame, the ball has an acceleration (-w^2r) toward the axis of rotation. The force, F_r, to cause this acceleration is supplied by the tethering cord, and we must have (In S) F_r=-mw^r

2. From the standpoint of a frame that rotates so as to keep exact pace with the ball, the acceleration of the ball is zero. We can maintain the validity of Newton's law in the rotating frame if, in addition to the force Fr, the ball experiences an inertial force F_i, equal and opposite to F_r, and so directed radially outward: (In S') F'_r=F_r+F_i=0, F_i=mw^2r.

In every single instance that the polar-coordinate concept is used, the term "centrifugal force" is only applied after transposing an acceleration term over to the force side of the equation. The question then becomes what this transposition corresponds to in terms of describing the physics and if it is different than transforming to a co-rotating frame? My reading of the sources, especially Taylor's Ch9, seems to indicate that the transposition includes an implicit transformation to a rotating frame. I will agree with you that a good deal of sources are not explicit in this, but there are sources, like French, that explicitly state the different reference frames without going through the full transformation equations. That is why there is a sentence explaining such at the end of the history section. As I've said before, lack of discussion of reference frames does not mean that we are still in an inertial frame. As Stommel indicates, the correct form of the radial acceleration in the inertial frame is ${\ddot {r}}-r{\dot {\theta }}^{2}$, not ${\ddot {r}}$. Therefore the only force in the inertial frame is F_r. --FyzixFighter (talk) 20:55, 1 June 2009 (UTC)

One more reference that does the same Goldstein 3-11 to 3-22 derivation and that explicitly states the reference frame transformation - "Introduction to Classical Mechanics" Atam P. Arya (1990), pg 231:

Making Eq. (7.56) [the radial component of F=ma for a central force problem with the r(theta-dot)^2 term transposed] look like Eq. (7.59) [mr-doubledot=F_eff(r)=F(r)+F_cent] has a a much deeper meaning. Using Eq. (7.59), we are observing the radial motion as viewed from a rotating reference frame. In this rotating frame, the force looks to be F_eff(r)

--FyzixFighter (talk) 21:25, 1 June 2009 (UTC)

As far as I can tell, your reason for wanting to suppress the "third meaning", even though it is presented in many reputable sources, and therefore cannot legitimately be suppressed, is that you believe this form of "centrifugal force" is actually an acceleration term that is brought over to the other side of the equation. Well, this is certainly true, but it is equally true for "first meaning" of centrifugal force. That's the whole point of inertial aka fictitious forces, they are really accelerations relative to rectilinear inertial coordinate systems, but when we use non-inertial coordinate systems (which may be stationary, if the spatial axes are curved) we get coordinate acceleration terms, which we can (if we wish) bring over to the other side and call "forces". The difference between the first and third meanings is simply that the first meaning handles curved time axes whereas the third meaning handles curved space axes.

I think the only legitimate way for you to argue against representing the views that I quoted from those references in the article is for you to somehow show that I falsified the quotes. Since I didn't falsify the quotes, I think that will be difficult for you to show. Those quotes are quite straightforward and unequivocal in describing centrifugal force in stationary curved coordinates, and I can provide more of the same. It isn't a terribly unusual point of view. It's actually more general and comprehensive view of fictitious forces than the more simplified view that you espouse.63.24.49.129 (talk) 02:20, 2 June 2009 (UTC)

So 130.-- and 63.-- are the same person - good to know. I agree with you that the first and the "third" meaning are really the same thing - which is why I don't see the need for a mentioning a distinct/independent "third" meaning. I disagree with your calling a curved coordinate system like polar coordinates a non-inertial coordinate system. That's one I haven't heard, do you have a reference that describes stationary polar-coordinates as a non-inertial frame?

IMO the important part of both of those references you provided is that they don't apply the name "centrifugal force" to the term until after they move the term over to the "force" side. Before the transposition, as you stated, the term is part of the acceleration in the inertial/stationary frame. The references you provided make no comment about whether the equation with the term transposed is descriptive of inertial or non-inertial frame. So again the question is does the transposing change observation frame. References like French and Arya and Taylor indicate that transposing the term from one side to the other physically corresponds to transforming from an inertial frame to a co-rotating frame. In other words by transposing the terms to the "force" side, you are no longer in a stationary curved coordinate system, but are in a rotating curved coordinate system. I would say the "first meaning" is more general than the "third meaning" - that is I can get the "third meaning" to pop out of the first meaning transformation by setting $\Omega ={\dot {\theta }}$ and having the axis of rotation be at the origin.

And please don't build a strawman argument out of some idea of falsifying quotes. I would never do that, and yet I still don't think your quotes are so unequivocal. Are you going to accuse me of falsifying the French and Arya quotes? That would be the easiest way for you to dismiss those quotes. I think those quotes plus others that I've been able to find are pretty unequivocal and straightforward that the "third meaning" is a subset of the "first meaning". --FyzixFighter (talk) 03:16, 2 June 2009 (UTC)

No, they are different meanings. Again, the "first meaning" takes coordinate accelerations due to curved TIME axes and treats them as fictitious forces, whereas the "third meaning" takes coordinate accelerations due to curved SPACE axes and treats them as fictitious forces. The thing about inertial coordinate systems being necessarily rectilinear (both spatially and temporally) is discussed in more sophisticated treatments of dynamics (e.g., Michael Friedmann's Foundations of Spacetime Theories), but often glossed over in more elementary treatments.

I can only repeat that I believe the references I quoted support the point I made, and the references you quoted do not contradict that point. Your idea that treating an acceleration term as a force is tantamount to adopting a rotating coordinate system is simply false, and is original reasearch. It applies only to acceleration terms that result from curvature of the time axis, not to acceleration terms that result from curvature of space axes, which is the "third meaning".130.76.32.182 (talk) 16:04, 2 June 2009 (UTC)

(unindent)The problem then is our difference on the interpretation of the quoted references, because I don't see anything in the references you quoted that contradict my point. I'm pretty sure where we disagree on the interpretation of the Stommel and McQuarrie quotes. I think we both agree that the equation of motion with the acceleration terms on the acceleration side of the equation "lives in" the stationary frame. The difference is then that we disagree in which "frame", stationary versus a rotating frame, the equation lives in after we transpose the acceleration term over to the other side. However, we do agree that the centrifugal force in the "third meaning" is still a fictitious force that does not appear in inertial frames (although we disagree what constitutes an inertial frame). Is that a fair summation of where we agree and disagree on the Stommel et alia references? I think we also disagree on the interpretation of the quotes I provided. IMO the Arya reference is especially clear in that it combines the first and third meanings. So can we step through at least the Arya reference and see where the point of divergence is? Arya starts by giving the polar-coordinate form of the radial component of F=ma in the stationary reference:

$m({\ddot {r}}-r{\dot {\theta }}^{2})=F(r)$

He later states that this can be rewritten as equation 7.56:

$m{\ddot {r}}=F(r)+r{\dot {\theta }}^{2}$

To me this looks exactly like the third meaning - taking an acceleration term in the polar-coordinate form over to the force side. Arya then identifies this term as the centrifugal force, gets rid of theta-dot using the angular momentum, and lumps it together with the applied force F(r) to give an "effective force", resulting in equation 7.59

$m{\ddot {r}}=F_{eff}(r)$

Again, up to this point I would say this is exactly the third meaning as I understand it. But then Arya makes the following statement:

Making Eq. (7.56) look like Eq. (7.59) has a a much deeper meaning. Using Eq. (7.59), we are observing the radial motion as viewed from a rotating reference frame. In this rotating frame, the force looks to be F_eff(r)

Which looks to me as the first meaning. Therefore, in my understanding Arya is treating the third meaning and the first meaning as the same concept. So at what point and how do we disagree about the Arya reference with respect to the first and third meanings being the same/different? Also, I'm unable to get a hold of "Foundations of Spacetime Theories" at the moment. Could you provide the relevant quotes from that source or another source, specifically addressing our disagreement about what constitutes? --FyzixFighter (talk) 18:59, 2 June 2009 (UTC)

You need to distinguish between frames and coordinate systems. The former are equivalence classes of the latter. In the more sophisticated treatments of the subject, these kinds of distinctions are crucial, and it is certainly key to understanding the difference between the "first" and "third" meaning.

I think your comments are mostly not relevant, because they are directed toward showing that SOMEONE doesn't recongize the third meaning, which isn't what you need to show. Everyone agrees that SOME people only recognize the first meaning, or the first and second. That isn't at issue. The point is that some people (i.e., reputable verifiable sources) recognize the "third" meaning, which is centrifugal force arising in stationary curved coordinate systems. Unambiguous quotes and citations have been provided in support of this. As I said before, unless you wish to claim that those quotes are falsified, there really is nothing to argue about. The third meaning needs to be represented in the article, per Wikipedia policy.

The arguement you've offered is the rather silly claim that whenever an acceleration term is treated as a force, we somehow magically turn the stationary coordinate system into a rotating coordinate system. If you want to include that (imho preposterous) point of view in the article, you're free to propose it, but I think there is very little support for it in the published scientific literature (for the simple reason that it makes no sense).130.76.32.182 (talk) 21:51, 2 June 2009 (UTC)

I should add that many reputable sources refuse to recognize the legitimacy of either the first or the third meanings of "centrifugal force", some going so far as to call it an abomination to ever refer to accelerations as forces. This too should probably be noted in the article. I can provide references and quotes if needed.130.76.32.182 (talk) 22:20, 2 June 2009 (UTC)

Wow, so no specific response to my attempt to determine where we disagree with respect to a specific source then. Stop acting so damn condescending and actually join in on the specific discussion, rather than simply dismissing my arguments as silly. That doesn't help come to a consensus. I've gone through the quotes you brought to the table and pointed out where we disagree in interpreting them. Can you give me the same courtesy?

You misunderstand my argument. I'm not saying that the "third meaning" is not found in the literature. I am saying that the literature is not clear that the third meaning and first meaning are distinct and different. I don't see how the Stommel reference is saying this, especially in light of some of the other statements made in Stommel. On the other hand I have found references that treat the first and third meanings as the same concept. I believe that the Arya reference is one such source that does this. Do you disagree? If so, where in my above stated interpretation of Arya do we disagree? And again, can you provide a quote for the statement that polar-coordinates are not an inertial frame so that I can at least better my understanding?

Finally, a source that clearly states that the first and third are distinct would help. This is especially important when there are references that specifically limit centrifugal force to two concepts. For example, from the Roche reference:

I have identified at least three interpretations of centrifugal force in the literature: a valid meaning in physics, an entirely different but equally valid meaning in engineering, and a cluster of false meanings.

The engineering meaning is the second meaning, the reactive centrifugal force that I believe we agree on. For the physics meaning, he states:

There is, however, a valid concept of centrifugal force in physics. If the observer in a frame of reference rotating with the Earth pretends for mathematical convenience that it is an inertial frame, then it becomes necessary to postulate a fictitious outwards force on a geostationary satellite to explain why it does not plunge to Earth. This is the centrifugal force of physics, an entirely fictional force.

--FyzixFighter (talk) 22:42, 2 June 2009 (UTC)

Along the same lines as the Arya reference, Tatum's "Celestial Mechanics" seems to treat the first and third meanings as the same concept:

Recalling the formula ${\ddot {r}}-r{\dot {\theta }}^{2}$ for acceleration in polar coordinates (the second term being the centripetal acceleration), we see that the equation of motion is

$m{\ddot {r}}-mr{\dot {\theta }}^{2}=F(r)$. (16.2.1)

This describes, in polar coordinates, two-dimensional motion in a plane. But since there are no transverse forces, the angular momentum $mr^{2}\theta$ is constant and equal to L, say. Thus we can write equation 16.2.1 as

$m{\ddot {r}}=F(r)+{\frac {L^{2}}{mr^{3}}}$. (16.2.2)

This has reduced it to a one-dimensional equation; that is, we are describing, relative to a co-rotating frame, how the distance of the particle from the centre of attraction (or repulsion) varies with time.

So again, a reference that does the third meaning technique of transposing the centripetal acceleration term, but interprets it within the framework of the first meaning. Am I misinterpreting these references? --FyzixFighter (talk) 22:55, 2 June 2009 (UTC)

This might be a useful reference for this discussion (and probably also for the article itself): "Donato Bini; Paolo Carini; Robert T Jantzen (1997). "The intrinsic derivative and centrifugal forces in general relativity: I. Theoretical foundations". International Journal of Modern Physics D. 6: 1. arXiv:gr-qc/0106013. doi:10.1142/S0218271897000029."

The main body of the text is mainly concerned with how the concept of centrifugal force can be given meaning in the general contect of GR, which is somewhat beyond the scope of this article. The first few pages, however, give a nice introduction which connects the usual ways in which centrifugal force is given meaning in non relativistic physics to the language of GR. (he actual identifies three different ways in which centrifugal force is used. The usual rotating frame one; one where a frame is comoving with some object such as a car and the fictious forces induced by the accelartion of the frame orthogonal to the motion of the frame are identified (the oscillating circle idea featuring in earlier versions of this article). And finally, the one in polar coordinates. He is quite explict that there is no rotation involved and that the term is due to the curvature of the coordinate system. He also show how this notion of centrifugal force can be connected to usual rotating frame one by going to a corotating frame. (TimothyRias (talk) 10:35, 3 June 2009 (UTC))

Yes, page 5 of that paper gives a clear statement of the meaning of centrifugal force arising from spatially curved but stationary coordinates, and it refers to other papers describing the same thing. This is consistent with the other reputable references that have been cited and quoted, and ought to be represented in the article, along with the meaning based on the co-moving frame.Lagu2 (talk) 17:36, 3 June 2009 (UTC)

Thanks for the reference, Tim. It certainly helps clarify the distinctions and connections between the different contexts. Now that we have a clear reference, I think we can move toward a equitable consensus based on all the reliable sources. Unless people object, I'd like to start a subsection where we can try to flesh a consensus text that most of us, if not all, can agree on. --FyzixFighter (talk) 18:16, 3 June 2009 (UTC)

FyzixFighter, I could answer that one for you. (1) What is the purpose of the imaginary co-rotating frame in the context of the two body planetary orbital problem, and how would we strap a co-rotating frame around a three body problem?

(2) Many textbooks do the central force problem using polar coordinates in the inertial frame without any mention of rotating frames. If you think that the co-rotating frame is equivalent to this mathematically, which indeed it is, then you are accepting that the inertial forces are relative to the inertial frame and not relative to the rotating frame.

(3) The rotating frame approach is aimed at dealing with non-co-rotating apparent deflections. The maths has gone completely off the rails for non-co-rotating situations, and we end up with nonsense concepts such as a 'radial Coriolis force'.

(4) In the polar coordinate approach, the Coriolis force is firmly fixed in the transverse direction.

Your best argument is that the similar mathematics between the polar coordinate approach and the co-rotating frame approach constitutes a kind of overlap between the two approaches. But an overlap is not the same thing as the two approaches being the same, such that you can completely erase one of the approaches from the article. We could then do the same with Newton's reactive approach. We could say that it is just the same for the special case of circular motion, and hence eliminate it from the article.

There are clearly at least three approaches to this topic, and Brews might be coming along soon with a fourth Lagrangian approach. You have got absolutely no business whatsoever trying to suppress the polar coordinates approach from the article. Your arguments of the kind such as 'just because rotating frames of reference aren't mentioned doesn't mean that they aren't there' simply don't wash with the wider public. You are effectively saying that because an imaginary concept such as a rotating frame of reference is not mentioned, doesn't mean that it isn't there, and hence it is an absolute essential, and so approaches that don't use rotating frames really are using rotating frames without knowing it. That just doesn't wash. David Tombe (talk) 23:36, 2 June 2009 (UTC)

@David: (1) See the section that starts on page 413 of Linton's "From Eudoxus to Einstein" where it wraps a rotating frame around the three body problem.

(2) Yes, essentially I am claiming that a transformation to a rotating frame is implicit in transposing an acceleration term to the force side of the the equation. The term that becomes the centrifugal force is, in the inertial frame, what some call the centripetal acceleration. I have yet to see a reference that calls it a centrifugal force without moving it over to the force side of the equation. I have to also yet so a reference that does not refer to the centrifugal force of either the first or third meanings as anything but "fictitious".

(3) That is your opinion, which also happens to disagree with modern physics sources for the last 100 years.

(4) This is only because the polar coordinate approach is tied to the corotating frame, see Taylor ch 9 and the Whiting reference

I seem to recall Symon's "Mechanics" page 365 explains the tie in between the Lagrangian "centrifugal force" and the rotating frame concept, unifying them, but I'll check tomorrow to see what it says exactly. And I don't think we can use overlap to eliminate the reactive centrifugal force. It's very distinct from the fictitious centrifugal force. The centrifugal force due to the rotation of the noninertial frame and the centripetal force act on the same point mass. The former force does not make a pair of action and reaction with any force, while the reaction of the latter force is the force exerted on the surroundings by the point mass.

@Anon130/63: Here's another quote from the Kobayashi paper:

The equation of the motion of a point mass is $m{\ddot {\mathbf {r} }}=F$, where m and ${\ddot {\mathbf {r} }}$ are the mass and the acceleration of the point mass, respectively, and F is the sum of the external forces exerted on the point mass. This equation can be expressed using plane polar coordinates (r, θ) of the point mass and the r-component and θ-component of the external force (F_r, F_θ) in the inertial frame S (figure 1). The r-equation is $m{\ddot {r}}-mr{\dot {\theta }}^{2}=F_{r}$ [3], where the second term on the left-hand side is the centripetal acceleration term (appendix A). The same situation can also be expressed in the noninertial frame S' which is turned relative to S. If the point mass moves along the x'-axis, the equation of motion with respect to S' is $m{\ddot {\mathbf {r} }}=F_{r}+mr{\dot {\theta }}^{2}$.

Later it states:

The reason the inertial force appears is that the frame has a rotational acceleration with respect to the inertial frame. If we write the inertial force with r and θ instead of r' and $\Theta$ [the rotating angle of S' relative to S], the origin of the inertial force is not clear.

If I can paraphrase my understanding of that last statement in relation to your disagreement, yes the third meaning uses coordinates that are defined in the inertial system, but expressing the centrifugal force in terms of those coordinates blurs the fact that the inertial force arises from a rotating frame. The Kobayashi also says something similar to Roche:

The term centrifugal force simply implies the force away from the center of rotation and is therefore obscure. Let us consider the point mass moving in a circular path with respect to the inertial frame. The term centrifugal force then has two meanings: one is the inertial force due to the rotation of the noninertial frame relative to the inertial frame and the other is the reaction force of the centripetal force to produce acceleration toward the center of rotation. The origins of these forces are different from each other.

--FyzixFighter (talk) 00:21, 3 June 2009 (UTC)

FyzixFighter, the $mr{\dot {\theta }}^{2}$ term in the polar coordinates expression is an inertial centripetal force. That same expression in Leibniz's equation (or equation 3-12 in Goldstein) is the centrifugal force. There is no question of a centripetal force ever becoming a centrifugal force simply by taking it to the other side of a specific equation. The Polar coordinate equation as when it is equated to zero for the inertial path is a different physical scenario from the Leibniz equation. You don't get from one to the other by taking terms across to the other side of the equation. They are different physical scenarios and the terms in each equation need to be analyzed accordingly. In the Leibniz equation, we have gravitational tails on the windward side of any radial motion, and the only centripetal force involved is gravity. In the inertial path equation, there is no gravitational field and we get an inertial centripetal force on the windward side of any radial motion, such as to cause a change in direction without affecting the speed. In practice, we could imagine this if we had a planar motion in the horizontal plane, in which case the gravitational tails would all be pointing upwards, and out of the way.

Rotating frames of reference only have meaning if they are real, such as a rotating platform. For example, if we have an object that is fixed on the platform, the glue or friction that fixes it there will provide the centripetal dragging force which will in turn be opposed by the inertial centrifugal force. The so-called rotating frame transformation equations focus on exactly such a co-rotating object. They do not actually cater for what they are claiming to cater for, which is the apparent deflections of non-co-rotating objects as viewed from a rotating frame.

As for planetary orbits, rotating frames simply don't come into it. There are no rotating frames of reference involved. All measurements are relative to the inertial frame. We use polar coordinates relative to the inertial frame and all the inertial forces are relative to the inertial frame. There are plenty of sources which deal with planetary orbits without having to corrupt the problem with this unnecessary political correctness about rotating frames of reference. So why are you so keen to insinuate that rotating frames of reference are an absolute necessity and to the extent that we shouldn't even be mentioning polar coordinates in the article at all. You are making out that you are 'Mr. Wikipedia Law keeper' and that you are only interested in seeing that all edits are properly sourced. But it is clear to all, that in truth you are only interested in sources which involve rotating frames of reference. Any sources which confirm the third way are summararily dismmissed by you, using specious arguments that are only likely to be swallowed by a hostile and politically correct audience. 10:05, 3 June 2009 (UTC) David Tombe (talk) 10:09, 3 June 2009 (UTC)

Moving toward a consensus text

Again, thanks to Tim for the reference. Maybe it would be best to begin with the points that we can all agree on and the points that will be the most common to the general readership. Would it be safe to say then that the uses of "centrifugal force" can be broken into two groups, one that uses the term for slightly different pseudo-/"fictitious"/inertial forces that only have meaning with respect to an implied frame of reference, and another that refers to the reactive (in terms of Newton's 3rd law) centrifugal force? I'd like to say that the reactive centrifugal force is independent of frame of reference and coordinate system. Is that an incorrect comment? In the first group, the centrifugal force is considered "fictitious" because it does not appear on the force side of the Newton's equation. Rather in the different contexts, it arises by transferring frame and coordinate-dependent acceleration terms to the force side of the equation. The three contexts are...and list the three context of the Bini paper. And then state how they all tie together, ie the first context being a special case of the second when the origin of the local frame is not accelerating. And the centrifugal force term of the polar coordinate context can be realized by adopting a frame that co-rotates with the particle and rotates about the center of force. We could move a lot of the discussion of the three contexts to summary section, and it might be better to rename the section "As a fictitious force" since rotating frames aren't exactly the concept that unifies the three contexts. Specifically at Lagu2 and Wolfkeeper, would something along these lines be agreeable? If not, where do we disagree, and do you have any suggestions for getting closer to something we can all agree on? --FyzixFighter (talk) 18:49, 3 June 2009 (UTC)

FyzixFighter, for a start, reactive centrifugal force is a special case of the polar coordinates centrifugal force. Imagine two bodies moving past each other in fly-by motion. According to Leibniz's equation (or 3-12 in Goldstein) there will be an outward centrifugal force acting between the two bodies along the radial line that joins them. If we then attach a string between the two bodies, this centrifugal force will pull the string taut. The tension induced in the string will then cause a centripetal force which will cause the two bodies to move in mutual circular motion.

Newton's centrifugal force is hence the special case of Leibniz's centrifugal force for circular motion. David Tombe (talk) 18:58, 3 June 2009 (UTC)

The characteristics of fictitious/pseudo forces is that you can't feel them; it's like gravity, if you fall off a roof you can't feel gravity- you only feel the ground when it hits you. The reactive centrifugal force is a real force that exists in all frames of reference, even inertial ones as well as non inertial ones. It is not the same as the centrifugal force in polar coordinates.- (User) Wolfkeeper (Talk) 21:35, 3 June 2009 (UTC)

That's important in accelerometers for example, accelerometers are completely insensitive to gravity, coriolis force and centrifugal force due to rotating frame of reference. They can feel reactive centrifugal force just fine!- (User) Wolfkeeper (Talk) 21:35, 3 June 2009 (UTC)

FF, yes, I think you have it exactly right, or almost so. The case of the non-reactive pseudo-force can be split up various ways: rotating coordinate system, system moving with the object, or system fixed at a center and co-rotating with object, and probably others; perhaps we need to find all we can, and describe them clearly as variations on a theme. Dicklyon (talk) 22:54, 3 June 2009 (UTC)

Wolfkeeper, it's true that you don't feel gravity until another force comes into play. Likewise with centrifugal force on most occasions. Mostly we only experience it passively. In the scenario which I described above, it becomes felt when it pulls the string taut. There is only one centrifugal force. Newton's reactive concept was a result of the fact that Newton was jealous of Leibniz's equation. He only invented the concept to be twisted. David Tombe (talk) 19:20, 4 June 2009 (UTC)

Nope. You directly measure reactive centrifugal forces just fine; it has immediate and very physical consequences. They are not the same thing as polar coordinate or rotating reference frame centrifugal forces or other pseudoforces such as gravity. IMO anyone that claims that they are the same thing is simply a crank.- (User) Wolfkeeper (Talk) 19:52, 4 June 2009 (UTC)

The General Polar Coordinate Equation and the Four Inertial Forces

This entire controversy can be solved by looking at the general polar coordinate equation in the inertial frame of reference. It takes the form,

${\frac {d^{2}\mathbf {r} }{dt^{2}}}=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}$

We have four terms on the right hand side. If we equate the right hand side to zero, we will have the inertial path, which is straight line motion. What do these four terms mean?

I'll begin by giving my opinion,

(1) ${\ddot {r}}$, "centrifugal force". It causes an outward radial expansion by virtue of changing the radial speed.

(2) $r{\dot {\theta }}^{2}$, "inertial centripetal force" which exists in the absence of a gravitational field tail eg. in a horizontal planar motion in which the gravitational field tails are all pointing upwards. It causes a change in direction without affecting the speed.

(3) $r{\ddot {\theta }}$, "transverse inertial force" which causes a change in the transverse speed.

(4) $2{\dot {r}}{\dot {\theta }}$ Coriolis force. A transverse force that causes a change in the direction of the radial motion. David Tombe (talk) 23:26, 1 June 2009 (UTC)

I'm puzzled. Why are you identifying these terms in the acceleration as forces? And why do you say that ${\ddot {r}}$ causes an outward radial expansion? I agree it represents, for the straight-line path, the fact that the path gets closer and then further. Is that what you mean by radial expansion? Of course, none of this explanation is particularly relevant if you can't point to a source that does so. Dicklyon (talk) 05:02, 2 June 2009 (UTC)

I don't really recognize the equation, although the RHS looks familiar. The RHS (right hand side) is the total acceleration of a body in a radial coord system, and the LHS is (or should be) F/m, the total force/mass which causes the acceleration. It is not dr^2/dt^2, but it is zero, if there are no "real" forces at work (gravity, EM, etc). The RHS is composed of our familiar radial and tangential pairs of accelerations. The r-hat is the outward radial unit vector and the theta-hat is the orthogonal "tangential" unit vector. If the LHS = 0 (for no force at all = inertial conditions) then each pair of accelerations on the RHS adds to zero also, since there is no radial or tangential acceleration, since there are no net radial or tangential forces. HOWEVER, that doesn't mean all individual terms on the RHS are zero. Just the pairs are. Thus ${\ddot {r}}=r{\dot {\theta }}^{2}$ for the radial Coriolis/centrifugal pair of accelerations (there is no centripetal one since there are no "real" forces) and $r{\ddot {\theta }}=2{\dot {r}}{\dot {\theta }}$ for the Euler and transverse Coriolis pair of accelerations. But even for staight line motion these all four terms have values for inertial motion. For example, the equation for an off axis line in polar coordinates is r(theta) = R_o secant (theta). If something is moving along that line with velocity V, then its y coords will increase by vt, and theta will be arcsin (vt/R_o). Differentiate that with regard to time for the above equation, and all kinds of nastiness results. However, one thing is clear, and it's that dr^2/dt^2 is NOT = 0. You can see that by just drawing the graph. Also d(theta)/dt is not constant, and it's second derivative isn't, either. SBHarris 07:16, 2 June 2009 (UTC)

The notation is somewhat confusing. The vector $\mathbf {r}$ is the displacement vector, its magntiude is the scalar $r$ and ${\hat {\mathbf {r} }}$ is a unit vector in the direction of $\mathbf {r}$, so

$\mathbf {r} =r{\hat {\mathbf {r} }}$

All three quantities are functions of time t, so we have

${\frac {d\mathbf {r} }{dt}}={\frac {dr}{dt}}{\hat {\mathbf {r} }}+r{\frac {d{\hat {\mathbf {r} }}}{dt}}$

and

${\frac {d^{2}\mathbf {r} }{dt^{2}}}={\frac {d^{2}r}{dt^{2}}}{\hat {\mathbf {r} }}+2{\frac {dr}{dt}}{\frac {d{\hat {\mathbf {r} }}}{dt}}+r{\frac {d^{2}{\hat {\mathbf {r} }}}{dt^{2}}}$

Now we have

${\frac {d{\hat {\mathbf {r} }}}{dt}}={\frac {d\theta }{dt}}{\hat {\boldsymbol {\theta }}}$

where ${\hat {\boldsymbol {\theta }}}$ is a unit vector perpendicular to ${\hat {\mathbf {r} }}$, and also

${\frac {d{\hat {\boldsymbol {\theta }}}}{dt}}=-{\frac {d\theta }{dt}}{\hat {\mathbf {r} }}$

so

${\frac {d^{2}{\hat {\mathbf {r} }}}{dt^{2}}}={\frac {d^{2}\theta }{dt^{2}}}{\hat {\boldsymbol {\theta }}}-\left({\frac {d\theta }{dt}}\right)^{2}{\hat {\mathbf {r} }}$

and finally

${\frac {d^{2}\mathbf {r} }{dt^{2}}}=\left({\frac {d^{2}r}{dt^{2}}}-r\left({\frac {d\theta }{dt}}\right)^{2}\right){\hat {\mathbf {r} }}+\left(r{\frac {d^{2}\theta }{dt^{2}}}+2{\frac {dr}{dt}}{\frac {d\theta }{dt}}\right){\hat {\boldsymbol {\theta }}}$

All of this is standard derivation of the acceleration vector in terms of 2-dimensional polar co-ordinates (for 3-d cylindrical co-ordinates, $\mathbf {r} =r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}$ and ${\hat {\mathbf {z} }}$ is constant, so just add a ${\frac {d^{2}z}{dt^{2}}}{\hat {\mathbf {z} }}$ term to the final result). No forces here at all, just components of the acceleration vector parallel to and perpendicular to $\mathbf {r}$ - as I have tried to explain to David on my talk page. Gandalf61 (talk) 10:18, 2 June 2009 (UTC)

OK, I've had three different responses. I copied the equation off Woodstone's talk page. The equation is not a matter of any doubt or controversy. I first saw it in my applied maths notes in 1979.

Gandalf61 has effectively derived it and re-written it in a slightly different notation. He has then stated his opinion that the four terms are not forces.

Dicklyon, as I predicted has capitalized on an irrelevancy. He has drawn attention to the fact that because the equation is kinematical without involving mass, that the terms are accelerations as opposed to forces. OK, so shall we then multiply the equation through by mass and then ask the question again? Why do we need to bother with these time wasting filibustering excercises? Dick has also tried to side track the issue once again to 'sources'. The 'sources' issue is over. We have sources. All we need now is to agree on the meanings of the four terms. So I see Dicks' reply as having been tantamount to a total ducking of the issue.

No, you miss the point. I understand that you can convert an acceleration to a force. But an individual term on the acceleration side is not usually identified with a force, esp. in polar coordinates where it doesn't correspond to any actual force. You effectively are treating the second derivative of the scalar r as if it were the whole acceleration when you do that, and that only makes sense in the co-rotating system, in which case it is identified with the centrifugal pseudo-force. Dicklyon (talk) 12:48, 2 June 2009 (UTC)

SBharris has carefully gone through the terms and made some correct observations. Unfortunately at the end, he informs us that the equation is not equal to zero. He has missed the point entirely. My question centred around the special case of when the equation is equal to zero, which of course makes it the equation for straight line motion. SBharris expresses his opinion that the four terms are not forces in the sense that he normally understands forces.

I think that we are all agreed that these four terms are not forces in the sense that we normally understand 'active forces'. They are inertial forces. They are built into straight line motion and they are quantified when we describe straight line motion in polar coordinates.

Now have a look at the section above and the argument between FyzixFighter and anonymous 63 -- -- --. FyzixFighter wants to suppress all references to the inertial forces in the article as they are treated in polar coordinates.

Sorry but no. I'm attempting to make the argument that the appearance of "centrifugal force" in polar coordinates is always accompanied by an implicit transformation to a rotating frame, ie transposing the centripetal acceleration term to the force side of the equation. See the Arya reference. Therefore, your "inertial forces" are a special case (the co-rotating frame) of the rotating reference frame concept. --FyzixFighter (talk) 15:12, 2 June 2009 (UTC)

That's your opinion. David Tombe (talk) 17:15, 2 June 2009 (UTC)

I would actually go further and say that this is the only truly correct way to treat the topic. Rotating frames of reference have got nothing to do with it. A rotating frame of reference merely adds and extra variable and causes an apparent circular motion to be superimposed on the inertial path.

The only relevance of rotating frames in all of this is when we have a physically real rotating frame in which constraining forces drag all the effects with the rotating frame. In that case, the inertial forces will be felt opposing the dragging forces.

In the popular literature regarding rotating frames of reference and fictitious forces, a transformation equation is derived which ostensibly caters for the apparent deflections. But in actual fact, that maths is just a re-run of the polar coordinates equation above and it deals with the inertial forces and not with the apparent deflections as it claims to be doing.

As to FyzixFighter's confusion as to why the literature calls the $r{\dot {\theta }}^{2}$ term centripetal force in the context of polar coordinates, whereas it becomes centrifugal force in Leibniz's equation, that can be explained. In Leibniz's equation, the entire physics changes. The gravitational field tails drop down into the horizontal plane and obliterate the inertial centripetal force. Gravity then becomes the centripetal force. The outward pressure in Leibniz's equation is then described by the term $r{\dot {\theta }}^{2}$ which is of course centrifugal force. The ${\ddot {r}}$ term which had been the centrifugal force in polar coordinates then becomes the general radial variable term in Leibniz's equation.

Again no, the literature calls it a "centripetal acceleration term" when it appears on the acceleration side. Big difference between acceleration terms and forces. There is not a required one-to-one correspondence between forces and terms in the acceleration. --FyzixFighter (talk) 15:12, 2 June 2009 (UTC)

FyzixFighter, I'm not going to play silly games of Force v. acceleration. David Tombe (talk) 17:15, 2 June 2009 (UTC)

That is why there was a conflict between myself and Brews. I wanted to give the centrifugal force a context within the planetary orbital equation, but Brews wanted to deal generally with polar coordinates. I argued with Brews that polar coordinates don't dictate a physical context in isolation. But thanks to Woodstone, I finally noted that when we equate the four terms to zero, we get the general equation for the inertial path in the absence of gravity and that we therefore expose four inertial forces as opposed to the commonly recognized three. David Tombe (talk) 12:19, 2 June 2009 (UTC)

You say:

I would actually go further and say that this is the only truly correct way to treat the topic. Rotating frames of reference have got nothing to do with it. A rotating frame of reference merely adds and extra variable and causes an apparent circular motion to be superimposed on the inertial path.

The problem I have with this, is that this is only your opinion, and there are no references to back this up, and indeed, there are huge numbers of references that disagree with this. The wikipedia does not ever, not ever include material because an editor believes that something is 'the only truly correct way to treat the topic'; it follows the literature. This is WP:NPOV, WP:VERIFIABILITY, WP:WEIGHT, it's absolutely fundamental. Even if you're right, it doesn't matter one jot. And you're not anyway; people need to do equations in rectilinear coordinates without constantly changing to and from polar coordinates.- (User) Wolfkeeper (Talk) 13:50, 2 June 2009 (UTC)

Wolfkeeper, I wasn't planning on putting my opinion about this into the main article. Neither was I planning on suppressing this approach to centrifugal force which I detest. You should be supporting me now regarding getting the polar coordinates approach into the article, because I know from your edits that you agree with me in substance if not in the details about this issue. David Tombe (talk) 17:13, 2 June 2009 (UTC)

David - Let me see if I understand all of this. I think you are saying that:

1. A particle moving at constant speed in a straight line relative to an inertial frame of reference is subject to four different inertial forces.
2. If we define a system of polar co-ordinates (fixed relative to the interial frame of reference) in a plane that contains the particle's path then, at any given time, two of these inertial forces act parallel to the vector $\mathbf {r}$ from the origin of this polar co-ordinates system to the particle and the other two inertial forces act perpendicular to $\mathbf {r}$ in the plane of the polar co-ordinates system.
3. For a particle moving at constant velocity relative to an inertial frame of reference, the pair of inertial forces parallel to $\mathbf {r}$ always nets to zero, and the pair of inertial forces perpendicular to $\mathbf {r}$ also always nets to zero.
4. We can place the origin of our polar co-ordinates system anywhere we like - there is no preferred location. When the particle is at a given point in its path, the direction and magnitude of each of these four inertial forces depends on where we choose to place the origin of our co-ordinate system.

Is that correct ? Gandalf61 (talk) 13:20, 2 June 2009 (UTC)

Gandalf61, Yes. That is exactly what I am saying. You have correctly identified that in either the transverse direction, or the radial direction, that the two forces cancel mathematically. But they don't cancel physically. In each case, one of the forces, ie. the Coriolis force and the inertial centripetal force cause only a change in direction but not a change in the speed. The other force ie. the centrifugal force and the other transverse force cause a change in the speed but not the direction. In the transverse case, this leads to the conservation of angular momentum.

If we now bring the gravitational tails down into the plane of motion and consider a two-body orbit, the force of gravity replaces the inertial centripetal force. We then have a Keplerian orbit and the conservation of angular momentum is Kepler's second law of planetary motion. Leibniz's equation,

${\ddot {r}}=-k/r^{2}+l^{2}/r^{3}$

describes the radial aspect of the planetary orbit and it solves to yield ellipses, parabolae, or hyperbolae. Kepler's second law can be used to convert the outward centrifugal term into an inverse cube law force of repulsion. David Tombe (talk) 17:07, 2 June 2009 (UTC)

David - good - I am glad I understood you correctly. I see now that you are boldy developing your very own special and highly original brand of Tombean dynamics. Good luck with that. Let us know when you publish. Gandalf61 (talk) 19:50, 2 June 2009 (UTC)

Yes, Tombe does understand polar fairly well you will indeed get weird forces when something moves in a straight line as seen from polar coordinates. Trouble is, he doesn't understand generalised rotating frames of reference (of which polar is a special case where coriolis is always transverse) at all (except for the special case).- (User) Wolfkeeper (Talk) 00:45, 3 June 2009 (UTC)

I see a good fit between the terms in the second deriviative and the rotational frame approach. The rotation $\mathbf {\Omega }$ is set equal to ${\dot {\theta }}$ times a unit vector in z-direction. By starting the right way, we keep the moving object on the $\theta =0$ line. In that frame the only observed movement is a one-dimensional ${\dot {r}}$ Then we have (leaving the mass m aside):

• ${\ddot {r}}$ corresponds to the acceleration in the rotational frame

• centrifugal force $-\mathbf {\Omega } \times (\mathbf {\Omega } \times \mathbf {r} )=-\Omega ^{2}r({\hat {\mathbf {z} }}\times ({\hat {\mathbf {z} }}\times {\hat {\mathbf {r} }}))=+{\dot {\theta }}^{2}r\,{\hat {\mathbf {r} }}$

• Coriolis force $-2\,\mathbf {\Omega } \times {\dot {\mathbf {r} }}=-2\Omega {\dot {r}}({\hat {\mathbf {z} }}\times {\hat {\mathbf {r} }})=-2\,{\dot {\theta }}{\dot {r}}\,{\hat {\boldsymbol {\theta }}}$

• Euler force $-{\dot {\mathbf {\Omega } }}\times \mathbf {r} ={\dot {\Omega }}r({\hat {\mathbf {z} }}\times {\hat {\mathbf {r} }})=-{\ddot {\theta }}r\,{\hat {\boldsymbol {\theta }}}$

−Woodstone (talk) 20:07, 2 June 2009 (UTC)

Woodstone, yes of course the two sets of maths are the same. That's been my whole point all along. But in neither case does the maths in anyway relate to rotating frames of reference. In the approach which you have just described, which ostensibly relates to rotating frames of reference, you have focused on a co-rotating object which is being dragged with the rotating frame. The actual inertial terms which you have derived are relative to the inertial frame. The centrifugal term has the angular speed of the object itself, and that is what is causing the centrifugal force. Your rotating frame is an imaginary concept which has got no bearing whatsoever on the physical reality of the situation.

No, they're not the same, polar coordinates is a strict subset and far less general.- (User) Wolfkeeper (Talk) 00:45, 3 June 2009 (UTC)

If we are interested in the apparent deflection of a non-co-rotating object as viewed from a rotating frame of reference, these inertial terms will be nothing to do with describing the superimposed apparent circular motion that we would observe.

Rotating frames of reference are only relevant to this topic if they are physically real, such as a rotating turntable. In that case, if we were to drag a radial motion on a groove, or drag a stationary object by friction or glue, or whatever, then we would feel the inertial forces pushing against the dragging forces. But those inertial forces are not referenced to the rotating frame.

So we are agreed that the two sets of maths are identical. But the two approaches in which this maths appears are quite different. One approach, which is the one used when we do central force problems, does not mention rotating frames of reference. It deals purely with polar coordinates relative to the inertial frame. The other approach is 'rotating frames' in the face, and it attempts to use the inertial forces to explain the apparent deflections of non-co-rotating objects as viewed from rotating frames of reference. There is absolutely no question of the two approaches being the same.

What we are up against here is are a couple of editors who have learned about centrifugal force within the context of rotating frames of reference and they have decided that this could be the only possible way to do it, and that the readers will not on any account be allowed to know about any alternative approaches, whether sourced or unsourced. David Tombe (talk) 22:39, 2 June 2009 (UTC)

The wikipedia is verifiability over truth, so unsourced is no good at all.- (User) Wolfkeeper (Talk) 00:45, 3 June 2009 (UTC)

Wolfkeeper, we've got sources. This is not about sources. This is about political correctness. David Tombe (talk) 10:06, 3 June 2009 (UTC)

Can you be specific about what statement of yours was objected to even though it was supported by a reliable source? Dicklyon (talk) 13:39, 3 June 2009 (UTC)

Dick, I'm not sure that we need to bother, because Wolfkeeper changes his position according to who he is arguing with. Recently he was arguing against an editor called Fugal to the extent that the polar coordinates approach is so different from the rotating frames approach that it is not even the same centrifugal force, and that it should have a page of it's own. Now he has decided to join ranks with FyzixFighter, after having locked horns with FyzixFighter only a few days ago on this same issue, and he is saying that the polar coordinates approach is a special case of the rotating frames approach. There is no point in presenting evidential exhibits to a kangaroo court. David Tombe (talk) 17:47, 3 June 2009 (UTC)

The rotating reference frame approach is considerably more general; the polar form is a subset. The thing rotating is different; in one case it's the frame, in the other it's an object (and a single object at that.) The centrifugal force equation looks the same, but the omegas refer to different things.- (User) Wolfkeeper (Talk) 03:01, 4 June 2009 (UTC)

Wolfkeeper, the co-rotating frame situation is a special case of polar coordinates when we have a physically real rotating platform. The inertial forces, which are all measured relative to the inertial frame are felt to be pushing against the dragging forces. In planetary orbits we don't have a rotating frame at all. Nothing is being measured relative to any rotating frame of reference in planetary orbits. All the inertial forces are being measured relative to the inertial frame. That's why the Coriolis force can contribute to stability nodes (Lagrange points) in the multi-body problems.

It's easy to build a physical rotating reference frame, people do it all the time. It's just a turntable. And note that in planetary orbits the mean anomaly constitutes a rotating reference frame, that is not aligned with the orbiting body.- (User) Wolfkeeper (Talk) 13:59, 4 June 2009 (UTC)

The apparent deflections that are observed on a non-co-rotating object from a rotating frame of reference can hardly be used to explain real stability nodes. Hence the rotating frames of reference approach is a fictitious approach which is quite different from the polar coordinates approach that is used in planetary orbital theory. Textbooks treat them as different topics and therefore so should wikipedia. And you know this fine well because your previous edits indicate so. David Tombe (talk) 10:04, 4 June 2009 (UTC)

To Editor Lagu2

I may be wrong, but I suspect that you are pro-relativity with a perhaps not entirely conventional approach, and that you have a particular interest in general relativity. So despite the fact that you are essentially saying the same thing as me regarding polar coordinates, that on more important issues we are perhaps diametrically opposed.

This is nevertheless interesting, because recently I discovered that my favoured approach to centrifugal force is in fact the approach that was first formulated by Leibniz. I have also heard it suggested that Einstein had two faces, and that his 'behind the scenes' face very much corresponded to that of Leibniz, whereas his public face was that of his theories of relativity.

I have also corresponded with people over the years who have been opposed to special relativity but in favour of general relativity. And I have also heard it suggested that general relativity can become a perfect theory if we take the special relativity virus out of it. I don't know what your views are on all of this, but I suspect that you are in favour of general relativity, or perhaps some variation of it. I can certainly see a link between general relativity and hydrodynamics, but I am of the school of thought that would want to totally purge it all of STR before taking it seriously. I am a believer in the fact that centrifugal force is a pure hydrodynamical force. And that may be the reason for the common ground between us as regards the preference for polar coordinates which are not related to rotating frames. Some editor last autumn who used a 63 IP server took a 'sit on the fence' approach between myself and the crowd, claiming that while I was anti-relativity, and wrong for being so, that my approach to centrifugal force tied in better with the relativistic approach, whereas the crowd, who are pro-relativity, had got it all wrong as far as centrifugal force was concerned. This 63 IP served made the comment that this distinguishes the half wits from the half wise.

Some say that Einstein was working behind the scenes in his later years on Leibniz and Boscovich. If you and Timothy Rias feel that this article needs to have a section on general relativistic centrifugal force, I would not oppose the inclusion of such a section, providing that it came below the polar coordinates and classical planetary orbital sections (which haven't been written yet). I doubt very much if I would agree with the contents, but I certainly wouldn't delete it, and I would read it with great interest. If you do so, please don't make it too mathematical because I never could grasp tensors. David Tombe (talk) 18:10, 3 June 2009 (UTC)

Third meaning

Some statements from earlier discussion:

The third meaning is a fictitious force in stationary but curved coordinates. Quite some time ago this was discussed here, and numerous references from reputable sources were provided for this definition, but all this seems to have been expunged from the article. I think, for accuracy and completeness, this third definition should be restored.63.24.42.134 (talk) 00:26, 31 May 2009 (UTC)

No, they are different meanings. Again, the "first meaning" takes coordinate accelerations due to curved TIME axes and treats them as fictitious forces, whereas the "third meaning" takes coordinate accelerations due to curved SPACE axes and treats them as fictitious forces.130.76.32.182 (talk) 16:04, 2 June 2009 (UTC)

Yes, page 5 of that paper gives a clear statement of the meaning of centrifugal force arising from spatially curved but stationary coordinates, and it refers to other papers describing the same thing. This is consistent with the other reputable references that have been cited and quoted, and ought to be represented in the article, along with the meaning based on the co-moving frame.Lagu2 (talk) 17:36, 3 June 2009 (UTC)

IMO this topic has validity and is indeed a third viewpoint. It is, however, outside the Newtonian view of mechanics, and the notions of "real" and "fictitious" forces. The real origin of this third topic is the Lagrangian-Hamiltonian formulation of mechanics, which is not based upon vector forces and vector accelerations and instead uses the scalar properties of kinetic and potential energy. A beautiful description can be found in C Lanczos. The Variational Principles of Mechanics. ISBN 0486650677..

The underlying terminology in the Lagrangian-Hamiltonian formulation is the "generalized coordinate" and the "generalized force" and for practitioners in this arena the word "generalized" often is dropped because the audience has already made the focus upon the variational approach, and the notions of "real" vs. "fictitious" forces never enter their minds. The issue for the article is how to introduce this viewpoint without adding total confusion, because the two views use the same vocabulary for incommensurable concepts.

Of course, from the Lagrangian-Hamiltonian viewpoint, r and θ are just some convenient coordinates and more generally the Jacobi coordinates can be used for a multi-body situation. The entire discussion now takes place in a manifold with its own metric and all that stuff and "forces" due to "space curvature" can be seen in that very mathematical context that greatly generalizes the simple use of polar coordinates, which becomes simply a pedestrian example. ("Space" may now have an arbitrary number of dimensions, greatly exceeding 3 or even 4.) An abbreviated introduction may be found at Mechanics_of_planar_particle_motion#Lagrangian_approach. In particular, Hildebrand's approach to polar coordinates from the Lagrangian viewpoint may be found there.

Here's a quote illustrating the use of terminology outside the Newtonian view:

In the above [Lagrange-Euler] equations, there are three types of terms. The first involves the second derivative of the generalized co-ordinates. The second is quadratic in $\mathbf {\dot {q}}$ where the coefficients may depend on $\mathbf {q}$. These are further classified into two types. Terms involving a product of the type ${{\dot {q}}_{i}}^{2}$ are called centrifugal forces while those involving a product of the type ${\dot {q}}_{i}{\dot {q}}_{j}$ for i ≠ j are called Coriolis forces. The third type is functions of $\mathbf {q}$ only and are called gravitational forces.

— Shuzhi S. Ge, Tong Heng Lee & Christopher John Harris: Adaptive Neural Network Control of Robotic Manipulators, pp. 47-48

Brews ohare (talk) 18:28, 3 June 2009 (UTC)

Brews, this sounds to me like a fourth meaning. I don't know about it, but I'd be interested to learn. So by all means write a new section on it. It should come after the polar coordinates section (which hasn't been written yet) and before a fifth section on relativistic centrifugal force. I personally never liked Lagrangian mechanics because it loses all track of cause. It is a kind of book keeping excercise with conservation of energy. If you write such a section, I may have something to add about grad(A.v) for the Lorentz force. That'll bring us full circle. David Tombe (talk) 18:51, 3 June 2009 (UTC)

If you look at the exposition in Francis Begnaud Hildebrand (1992). Methods of Applied Mathematics (Reprint of 2nd Edition of 1965 ed.). Courier Dover Publications. p. 156. ISBN 0486670023., you will see it has much in common with Goldstein. (I'd be inclined to say they are identical if you look at Goldstein's book in the large, which is a Lagrangian formulation). The general relativity approach also falls under this rubric, where force is related to metric. See Walecka & Koks & Ortin.Brews ohare (talk) 18:59, 3 June 2009 (UTC)

I found this quote that draws a connection between the Lagrangian centrifugal force and the Newtonian "fictitious" centrifugal force.

...the equation of motion for $q_{k}=r$ is $m{\ddot {r}}=F_{r}+mr{\dot {\theta }}^{2}$. If we compare this with Eq. (3.207), which results from a a direct application of Newton's law of motion, we see that the term $\partial T/\partial r$ is part of the mass times acceleration which appears here transposed to the right side of the equation. In fact, $\partial T/\partial r$ is the "centrifugal force" which must be added in order to write the equation of motion for r in the form of Newton's equation for motion in a straight line. Had we been a bit more clever originally, we should have expected that some such term might have to be included. We may call $\partial T/\partial q_{k}$ a "fictitious force" which appears if the kinetic energy depends on the coordinate q_k. This will be the case when the coordinate system involves "curved" coordinates, that is, if constant generalized velocities ${\dot {q}}_{k},...,{\dot {q}}_{3N}$ result in curved motions of some parts of the mechanical system.

— Keith R. Symon: Mechanics (1971), pg. 365

--FyzixFighter (talk) 19:33, 3 June 2009 (UTC)

Hi FyzixFighter: This is an example of an author trying to connect these different worlds, as does Hildebrand, linked above. A question in my mind is whether this author has succeeded, or has simply reiterated the confusion. It may be better to treat the two approaches as distinct and let the reader draw any connections for themselves. In particular, the basic Lagrangian formulation is much more general than this comparison, and reduces to this comparison only for this case. It seems better to me to view both approaches as separate, rather than seemingly imply a basic connection that, in fact, is not basic at all, but exists only for a very specific narrow example. Brews ohare (talk) 19:47, 3 June 2009 (UTC) I've taken a stab at such a section. Brews ohare (talk) 22:16, 3 June 2009 (UTC)

Brews, I don't see how your cited source supports your statement that The Lagrangian use of "centrifugal force" has a very limited connection to the Newtonian definition, a connection applicable only in the very particular case of polar coordinates. It arrives at the same result, a fictitious force due to the rotating coordinate system. Sure, it's a more general approach, but to the extent that it is used to arrive at centrifugal force, it's the same centrifugal force. Maybe I don't know what you mean by Newtonian here; I think we shouldn't use that term, as sources clearly indicate that Newton flipped between different meanings of CF. Dicklyon (talk) 01:20, 4 June 2009 (UTC)

Hi Dick: Read the quote: "Terms involving a product of the type ${{\dot {q}}_{i}}^{2}$ are called centrifugal forces" Is that what you are talking about? There need not be any rotation, the q_k are not radial, and there may be 100 of them. Their physical interpretation is all over the map. Read some of the references and some of the associated wiki articles.Brews ohare (talk) 06:33, 4 June 2009 (UTC)

So it's not at all unrelated, just more general to include additional possible forces besides the usual one called centrifugal force. That's not a "very limited connection", but a generalized definition. Dicklyon (talk) 14:03, 4 June 2009 (UTC)

Hi Dick: A "generalized centrifugal force" is not a Newtonian force. It does not transform as a vector, for example. In the very particular case of a body where the "generalized coordinates" are ( r, θ ) the link to Hildebrand shows the Lagrangian approach, and it leads to the same equations for $({\ddot {r}},\ {\ddot {\theta }})$ found in a co-rotating frame using Newton's laws with fictitious forces. (Of course, deriving the same equations does not mean that interpreting the "generalized acceleration" ${\ddot {r}}$ as a Newtonian acceleration in an inertial frame is appropriate (as you know)). Because mechanics is mechanics, whether it is done using Newtonian forces or a Lagrangian, the equations found either way must be the same when using the same coordinates as variables. This connection may be very convoluted however, and the Lagrangian variables may not relate simply and directly to one unique frame of reference. (For example, rotations about joints centered at different articulation points themselves in relative motion.) There is no necessity for the terms identified as "centrifugal" in the Lagrangian approach to be also Newtonian forces in some frame, it is not an objective of the Lagrangian approach: many examples of "generalized forces" have no such connection, and need not even have units of newtons. The references to robotic design are replete with such examples. As a rule, no attempt is made in Lagrangian mechanics to derive the equations of motion from the Newtonian framework to see in what manner it can be contorted to find how it leads to the same equations, and no-one is concerned that "generalized forces" are not "Newtonian forces". Brews ohare (talk) 17:25, 4 June 2009 (UTC)

I'd go so far as to say the Lagrangian approach makes use neither intuitively nor mathematically of any supposed relation of the "generalized centrifugal forces" and the "generalized Coriolis force" to the Newtonian objects of the same name. These names are no more than labels used to refer to the terms when discussing manipulations. The equations usually are arranged for application of standard numerical methods, and the separation of the double-dot terms on the left of the equations and grouping the other terms on the right according to their derivative structure (whatever names you attach to these terms) makes numerical solution a standard operating procedure. Calling them "generalized forces" in the field of robotic design is nothing more than a picturesque parachronism. Brews ohare (talk) 17:36, 4 June 2009 (UTC)

Lorentz force

Brews, just as an aside, if you go to the Lagrangian chapter in Goldstein, you will see that he attempts to derive a Lagrangian for the Lorentz force. He ends up pulling out A.v as the Lagrangian for the vXB term. Now have a look at this Eric Weisstein link on the Lorentz force and note in particular how he has converted vXB to grad(A.v) at equation (8) [5]. Now you are aware that Dirac linked A to velocity and Maxwell linked it to momentum. Hence A.v is that very same kinetic energy term that is used as the centrifugal potential energy in your rotating bucket argument. And this ties in with Maxwell's attempt to explain vXB as the force on a current carrying wire, in terms of centrifugal pressure coming from his molecular vortices, in which the vorticity is measured by B. Does this help you to see how centrifugal force is a relative quantity in the sense that it has a different value and direction according to which arbitrarily chosen point in space that it is measured relative to (a fact pointed out by Dan Bernoulli), yet it is absolute in that, having chosen a point, the value will be dependent on the absolute rotation relative to the background stars? Does this not point to a 'wheely' nature for space. An inverse cube law dipole field in which pressure is generated by angular acceleration. Do you not yet see how it all ties up? Can't you see the tiny Archimedes' screws in the very engine rooms of space?

And I should further add that your point about the radial direction is correct. When the truth comes out and the situation is generalized, the centrifugal force will not be confined to act in the radial direction. There can be a transverse effect, and also an axial effect (as in gyroscopes that are precessing). Gryoscopic precession that can defy gravity is a totally misunderstood aspect of the inertial path. These compound centrifugal forces are totally misunderstood. The transverse compound centrifugal force is known as the Coriolis force in polar coordinates. David Tombe (talk) 10:18, 4 June 2009 (UTC)

Removing edits from the talk page

Wolfkeeper, you have done absolutely nothing to help this article. All you have done is taken whatever inconsistent position is necessary to be in disagreement with me. Only a few days ago you were saying that polar coordinates are a different approach to centrifugal force than rotating frames. Then you changed your position to the fact that one was a special case of the other.

And now you are removing edits from the discussion page. I was introducing the idea of a section on relativistic centrifugal force to see what the attitude was. It is already now being discussed. Are you just looking for an edit war? David Tombe (talk) 18:59, 4 June 2009 (UTC)

Stick to the topic or leave. One or the other.- (User) Wolfkeeper (Talk) 19:43, 4 June 2009 (UTC)

Wolfkeeper: You are becoming intemperate; removing material from the talk page because you find it beside the point is not your prerogative, is annoying, and interferes with discussion. Brews ohare (talk) 19:54, 4 June 2009 (UTC)

It was a message to a specific user and it was not on topic.- (User) Wolfkeeper (Talk) 19:56, 4 June 2009 (UTC)

It was a message to a specific user, and it was of interest to all concerned in the discussion, and it was totally related to improving the article. It introduced the idea of centrifugal force in general relativity and I'd like to hear Lagu2's views on the matter. I'm going to put it back again because you have no business removing it. It would seem that you are only trying to provoke a revert war on the talk page. David Tombe (talk) 21:08, 4 June 2009 (UTC)

This is simply abuse of your editing privileges. I can see a time coming soon where they will be taking away from you again.- (User) Wolfkeeper (Talk) 23:45, 4 June 2009 (UTC)

The Reactive Centrifugal Force

FyzixFighter stated above that the reactive centrifugal force applies in any frame of reference. He then asked if that is a correct comment. It's not a correct comment.

The reactive centrifugal force acts in the same frame of reference that the corresponding centripetal force acts in. That is the inertial frame of reference.

There is only one centrifugal force. But there are a number of approaches to the topic which do not fully agree with each other. David Tombe (talk) 19:14, 4 June 2009 (UTC)

I do not agree with David. The reactive centrifugal force is a real force from Newton's law of action and reaction (exerted upon the mechanism providing a centripetal force that maintains a body in its path), and as such transforms like a vector under coordinate changes and originates in a real identifiable source (namely, in the body following the path). Brews ohare (talk) 19:24, 4 June 2009 (UTC)

FyzixFighter is completely correct. You do not understand this point David.- (User) Wolfkeeper (Talk) 19:55, 4 June 2009 (UTC)

If you can find a source that supports your claim that the reactive centrifugal force is the same as the "fictitious" centrifugal force of rotating frames and polar coordinates then we can include it. All the references I've found that talk about the two are clear that they are different concepts. One of the main differences being that the centripetal and "fictitious" centrifugal force act on the same object, whereas the centripetal and reactive centrifugal force act on different objects. --FyzixFighter (talk) 20:27, 4 June 2009 (UTC)

I'm afraid FyzixFighter is correct. Although the two forces point in the same direction and have the same magnitude for circular motion (this making them easy to mix up!) indeed they act on different objects. The reactive force is just the other end of the "two-headed" action-reaction arrow of Newton's third law. It is seen in all frames, both accelerated and not, when looking at an accelerated object (think of the box on a spaceship). It acts on the ship (points toward the floor and pushes on the floor). Its other end is the centripetal force, which also appears in all frames. But the centrifugal force is an ADDITIONAL arrow without an end, which acts ONLY on the box, and is seen ONLY in the accelerated (ship) frame, and disappears otherwise. It doesn't follow Newton's third law. So yes, it's a different animal. SBHarris 20:36, 4 June 2009 (UTC)

No Steve, this is another case of you not having considered the whole picture and assuming that there is something that I have overlooked. I have not overlooked the fact that Newton's third law acts over two bodies. Read this reference, beginning near the bottom of page 268 [6], and also my reply below to FyzixFighter. David Tombe (talk) 20:55, 4 June 2009 (UTC)

I read the section and it's irrelevent. Yes, Newton was wrong that centrifugal force and centripetal/gravity are an action/reaction pair, since there are cases where they are not equal. It is centripetal/gravity and "reactive centrifugal" which are the Newton's third law action-reaction pair. But note that "reactive centrifugal" force is NOT the same as centrifugal force. The first is always the same magnitude (but opposite direction) as centripetal force, the second is not! That is merely one more argument that they should be treated differently. Besides the fact that one of them-- the plain centrifugal force-- can be made to disappear in a suitable frame, while the other one (the reactive centrifugal), if it exists, cannot be made to disappear in any frame. SBHarris 23:04, 4 June 2009 (UTC)

FyzixFighter, the 1961 Nelkon & Parker source which you erased was quite clear about the fact that the reactive centrifugal force and the centripetal force are acting on the same object.

I know that you will say that Newton's 3rd law of motion needs to act over two bodies. But the point here is that Newton's reactive centrifugal force is not compatible with his third law of motion. I have a good source which points this fact out. It is to be found in the history section. In Leibniz's equation, centrifugal force and gravity (the centripetal force) both act on the same object and they are not in general equal and opposite. They are both free to have different magnitudes and they are certainly not an action-reaction pair. Newton on seeing this equation simply decided to be twisted. He declared that centrifugal force is an equal and opposite reaction to centripetal force as per his third law of motion. The source also indicates that Newton didn't really believe what he was saying, and indeed anybody with any comprehension of the topic would know that Newton was talking nonsense on this issue.

Newton's concept does however have a modicum of truth in the special case of circular motion, in that the centripetal force and the centrifugal force are indeed equal and opposite. But they are not an action-reaction pair. The centrifugal force in such situations is the one and only centrifugal force which pushes outwards due to rotation and which is measured relative to the inertial frame of reference. It is the centrifugal force in Leibniz's equation, equation 3-12 in Goldstein, and the (r double dot) term in the polar coordinate equation for the inertial path.

Basically, you are engaged here in trying to enforce scientific political correctness, and the issue which offends you is the simple law of nature which tells us that spinning fluids expand outwards and push against their surroundings. That is what centrifugal force is all about, but you are offended by it, and so you want to make sure that it is written up here in a distorted fashion. You want to promote a rubbish idea, which is the popular political correctness of the day. You want everybody to believe that centrifugal force is only a product of how we look at something and that it is only an apparent deflection that is observed from a rotating frame of reference.

We all know that you have got millions of references to back that nonsense idea up. But what you are now doing now is making sure that all references to alternative approaches are suppressed. David Tombe (talk) 20:49, 4 June 2009 (UTC)

It has nothing to do with PC. It's just that there are two different conceptions of centrifugal force, one being the equal and opposite reaction force that acts on another object, the other being a pseudo force due to frame rotation. None of your sources have ever suggested any third alternative, or do they confuse these two as you do. Dicklyon (talk) 23:14, 4 June 2009 (UTC)

Dick and Steve, Can you then tell me where the polar coordinate centrifugal force gives way to the reactive centrifugal force in the scenario of the two spheres? Consider two spheres in mutual fly-by motion. An outward radial centrifugal force exists between them as per equation 3-12 of Goldstein, as per Leibniz, and it is an inverse cube law force. If we then attach a string between the two spheres, this centrifugal force will pull the string taut and hence induce a tension in the string.

Where does the polar coordinates centrifugal force differ from the so-called reactive centrifugal force such as to make it a different thing? David Tombe (talk) 03:30, 5 June 2009 (UTC)

When you analyse that scenario in polar coordinates, you only analyse one half of it; the other half is dealt with later due to symmetry.

Taking the half, there is a centripetal force due to the string which pulls the string towards the centre of rotation.- (User) Wolfkeeper (Talk) 04:05, 5 June 2009 (UTC)

Slightly less obviously there is a force on the other sphere that is a reactive centrifugal force. This force points in the opposite direction to the centripetal force, but has crossed the rotation centre. A force can be said to act anywhere along its line of action, so it's a perfectly good reactive centrifugal force. It is a real force.- (User) Wolfkeeper (Talk) 04:05, 5 June 2009 (UTC)

This is a slightly odd example though, often the object experiencing the reactive centrifugal force is not rotating; for example in a wall of death.- (User) Wolfkeeper (Talk) 04:05, 5 June 2009 (UTC)

David, I'm not aware of anything called "polar coordinates centrifugal force" or anything like that, so it's hard to answer that question. Centrifugal force doesn't depend on the type of coordinate system, just on the rotation of your reference frame; of course, polar coordinates make it easier to deal with rotating frames, so maybe that's what you mean. In that case, the difference is clear, for example in your favorite equation, where the reaction force to gravity is not necessarily equal to the pseudo-force known as centrifugal force. Your confusion is addressed specifically by Stommel and Moore. Dicklyon (talk) 04:09, 5 June 2009 (UTC)

And in any case, if the spheres are in linear "fly-by" if there is no string or gravitational interaction, all the forces on them are fictitious (products of coordinate system choice) and can be made to go away by choosing Cartesian coords. The fact that force-like terms show up in accelerated frames and polar coordinates only means that they are pseudoforces at best, not subject to Newton's third law, and aren't "real."

If you attach a string between the spheres, no real-force acts anywhere until the string IS taut, and then obviously you have an action-reaction Newtonian force-pair following the third law, and induced by the string. THAT force (string tention acting on both spheres) is the SAME in any coordinate system and any frame. It is real. But it is not "induced" by other forces. It is a primary force INDUCED by motion, like pulling on a spring by any means (including your hands). Don't put the cart before the horse, or reason for existance of a force before the force. A string does NOT cause force because you pull on its ends. Rather, you move its ends outward (forcelessly), and THEN the force comes when it hits its taut length limit and you continue the motion. Same with a rubber band. Motion first; force later. Two balls connected with a rubber band and moving outward illustrate this well-- no force until they hit the band limit, then force caused only by their linear motion away from each other, since after a point in their motion, they are trying to pull connected molecules apart by continuing their travel. SBHarris 18:23, 5 June 2009 (UTC)

Steve, Even before the string becomes taut, there is a centrifugal force as per Leibniz's equation. This is the force that pulls the string taut. If not, what force do you think pulls the string taut? David Tombe (talk) 09:28, 8 June 2009 (UTC)

Steve, Dick, and Wolfkeeper, The centripetal force only acts after the string has become taut. The string becomes taut because of the centrifugal force which you are claiming is only fictitious. I am not the one that is putting the cart before the horse. It is those who believe that reactive centrifugal force is a reaction to the centripetal force that are putting the cart before the horse.

Please read my comment again. Two balls or objects moving directly and inertially away from each other would experience the same problem, if connected by a string longer than the distance between them. As they continued their motion, the string could straighten out without exerting any force, due to the spacial motion (separation) of the objects. When they hit the limit, it would become taut and exert a force on both. There is no "centrifugal" here-- just simple motion and backward pull. This is an ideal string, of course, and any real string would take a tiny force to straighten, and would exert a very tiny drag force in return (by its very mass and own inertia if nothing else). But in the limit, these are zero, as they may be made as small a fraction as you like. You can connect two asteroids moving linearly apart by a slack thread: there is no appreciable force until the thread reaches its limit, and then the force grows to the limit of the mechanical strength of the thread. The linear inertial motion of an object in space does not exert any forces, require any forces, and generally is not the product of any forces. That's Newton's first law. SBHarris 22:28, 7 June 2009 (UTC)

As for Newton's 3rd law of motion, it acts over two bodies. We are not concerned with it in this example. We are focused on what happens at one end of the string.

I have asked you to explain precisely where this so-called fictitious centrifugal force differs from the reactive centrifugal force. I can only see one single centrifugal force in the example. There is the centrifugal force which pulls the string taut. Even before it started pulling on the string, it's presence was clearly accounted for in Goldstein's equation 3-12, or in Leibniz's equation. It is the radially outward inverse cube law term which you deem to be fictitious. David Tombe (talk) 14:36, 6 June 2009 (UTC)

In the case of the gravitationally attracting spheres, each centripetal force is also the reactive centrifugal force for the opposite sphere.- (User) Wolfkeeper (Talk) 15:55, 6 June 2009 (UTC)

Wolfkeeper, that would be a logical way to look at it. But Newton's reactive centrifugal force was not logical. He purported to be working from his third law of motion, but in actual fact, his appliaction of the concept was to the one single object. You guys have tried to write up Newton's concept and simultaneously correct it. I showed you a reference to a 1961 Nelkon & Parker which correctly stated Newton's faulty concept. You guys deleted it, and the source, and replaced it with what you thought Newton's definition really should be. David Tombe (talk) 14:40, 7 June 2009 (UTC)

At best, that's of historical interest only. The reactive centrifugal force and the centrifugal forces act in opposite directions in this example and in general they are completely different things, they are generated in different ways, and they usually apply to different objects.- (User) Wolfkeeper (Talk) 15:14, 7 June 2009 (UTC)

Wolfkeeper, There are four forces involved over the two objects. There are two pairs of action-reaction pairs. There is a pair of centripetal forces both equal and acting in opposite directions, and there is a pair of centrifugal forces both equal and acting in opposite directions. But all four forces are only equal in magnitude in the special case of circular motion. David Tombe (talk) 09:25, 8 June 2009 (UTC)

Let's see if I understand the Tombean dynamics here. Apparently, at some point the centrifugal force that was acting on the flying ball starts to act on the string instead, pulling it taut. Is that correct ? Or does it continue to act on the ball while it is also acting on the string ? Or is part of it acting on the ball, and part acting on the string ? And what about the other 3 inertial forces that are acting on the ball in Tombean dyanmics - why do they not also start acting on the string ? Does the centrigugal force have some special string-pulling property that the other Tombean forces lack ? Or is the string perhaps made of some material that is impervious to the other Tombean forces ? Gandalf61 (talk) 15:43, 6 June 2009 (UTC)

IMO the problem David must face is that he employs an intuitive view that switches unconsciously from a Newtonian force formulation to a curvilinear Lagrangian formulation, and applies the viewpoints interchangeably without realizing it. Brews ohare (talk) 19:42, 6 June 2009 (UTC)

Gandalf61, Of the four inertial forces, only the outward centrifugal force pulls on the string. The tension in the string is a radial effect and so the two transverse forces will have no effect on that tension. The inward inertial centripetal force only has the effect of changing the direction of the sphere and not the speed. So yes, only the centrifugal force will pull on the string.

Regarding your other questions, the centrifugal force acts on the sphere, as per the Leibniz equation (or equation 3-12 in Goldstein). There is then a knock-on effect in which the fact of the string being connected to the sphere results in the string getting pulled taut.

For simplicity, I'll switch to a 'push' situation and consider a wall of death rider. The centrifugal force acts on the motorbike. The motorike then transmits that effect to the wall, causing a pressure which induces a 'normal reaction'. The normal reaction causes a centripetal force. If the wall of death is circular, that centripetal force will be equal and opposite to the centrifugal force, but the two will not form an action-reaction pair. Nevertheless, Newton's concept of reactive centrifugal force is a faulty concept, based on his jealousy of Leibniz's equation, and so the centrifugal force pushing on the wall is indeed the so-called reactive centrifugal force as defined by Newton, even though in reality it is pro-active.

In the case of the wall of death, we can compare the centrifugal force to gravity in the sense that it acts on the motorbike, whereas we can compare the 'reactive centrifugal force' to 'weight', in that it is the knock-on pressure against a surface which induces the normal reaction.

Regarding Brews's comments, he is half correct. But I simply don't recognize the current theory that centrifugal force is an apparent force which is viewed from a rotating frame of reference. As far as I am concerned, if we view something from a rotating frame of reference that is not co-rotating, then we will have an apparent circular motion imposed on top of the already existing motion. That apparent deflection has got nothing to do with centrifugal force. I do not link this deflection with Newtonian mechanics in any respect.

Quite a few of the editors here have now realized that the inertial forces of polar coordinates are one and the same thing as those which occur in the co-rotating frame scenario. Correct. But it is the co-rotating frame that is a sub-set of the polar coordinate inertial forces. Co-rotating frames only exist on paper when they also exist in reality, such as in the case of a rotating turntable with dragging forces that drag a scenario, or in the case of the rotating Earth that entrains the atmosphere. But with planetary orbits, there is no rotating frame of reference involved, and the centrifugal force is a pure inertial force. Those modern authors who have decided to strap an imaginary rotating frame of reference around the Kepler problem need to ask themselves 'why?'. It means nothing and there is absolutely no need to do so. David Tombe (talk) 14:33, 7 June 2009 (UTC)

Your explanation "The tension in the string is a radial effect and so the two transverse forces will have no effect on that tension" assumes that the origin of my polar co-ordinates is somewhere along the line of the string. But we have established that in Tombean dynamics there are no preferred locations for the origin - I can place the origin of my polar co-ordinates anywhere I like. So I place my origin at a point that is not on the line of the string; in other words, the string is not radial in these co-ordinates. Now the "transverse" forces, which are perpendicular to the radius vector, are not perpendicular to the string. So do you have some other explanation why only the centrifugal force acts on the string ? Gandalf61 (talk) 15:19, 7 June 2009 (UTC)

Gandalf61, If we discuss a practical scenario such as two fly-by spheres with a string linking them, but then take our polar origin point to be in some other totally impractical place away from the two spheres, the situation will be much more difficult to analyze mathematically. The physics of course will not change. But the tension in the string, as measured using polar coordinates based on an impractical origin not along their mutual radial line, will then be due to Coriolis force and/or the other transverse force.

See my response to Brews below in the Lagrangian section. I had mentioned centrifugal pressure and the fact that grad(A.v) could never mean Coriolis pressure since the two velocity terms are mutually perpendicular in Coriolis force. However, Coriolis pressure does exist because the two transverse forces are pressing against each other. In the grad format, only irrotational pressure shows up. The transverse pressure is catered for under Lenz's law. If we shift the origin, transverse pressure can then show up as radial centrifugal pressure.

Kepler's second law and Faraday's law are essentially the same thing. They deal with the transverse forces and Coriolis pressure. David Tombe (talk) 09:17, 8 June 2009 (UTC)

Lagrangian section

User:Rracecarr commented out the section Centrifugal_force#Lagrangian_framework with the laconic comment that "No attempt is made to explain what is meant by "Lagrangian centrifugal force"". This remark requires further amplification inasmuch as the term "Lagrangian centrifugal force" is never used in the article and is not necessary. In its place is a very clear statement:

"Terms involving a product of the type ${{\dot {q}}_{i}}^{2}$ are called centrifugal forces"

which is not only clear but a direct quote from a reputable source.

I believe further discussion of these ideas is warranted here before engaging in deletion of well documented material. Such discussion should be cognizant of the section Talk:Centrifugal_force#Third_meaning. Brews ohare (talk) 17:33, 5 June 2009 (UTC)

As it stands, the section is terrible. Such a section could certainly be included, but it would need to start with some sort of understandable indication of what the supposed meaning of "centrifugal force" is in the context of Lagrangian dynamics, rather than simply the statement that it "employs the term", which is devoid of information. Also, the length of the section and its early placement in the article place entirely undue weight on this relatively obscure meaning. Third, the early parts of an article on a basic concept such as centrifugal force should be written to be accessible to a layperson. Diving into Lagrangian dynamics before any sort of attempt is made to explain intuitively what centrifugal force (in the most common usage) actually is, or any attempt to address the common misconceptions about it, is entirely inappropriate. Rracecarr (talk) 17:58, 5 June 2009 (UTC)

How would you approach introduction of Lagrangian mechanics "to be accessible to a layperson"? Why must the article be understandable in its entirety to a "layperson"? There are many counter examples in Wiki, in fact, in technical areas like this one they are very common. I find your standards are (i)unnecessary and (ii) arbitrary. I have rewritten the section, nonetheless. Brews ohare (talk) 18:42, 5 June 2009 (UTC)

It is worded somewhat better now, but still quite ridiculous in terms of undue weight. Take a look at the article in any of the other languages. Where is omega cross omega cross R? Do you really think it makes a good article to spend a couple hundred words, in total, on the "everyday", "fictitious", and "reactive" meanings, and then a couple hundred more on "Lagrangian"? It is out of all proportion.Rracecarr (talk) 18:53, 5 June 2009 (UTC)

On a different note, this topic has recurred over and over again since I began looking at Centrifugal force, and it is not going away. It engages this talk page for months at times. Some means of dealing with it is needed, and its usage is not exactly obscure: besides polar coordinates and robotics, the entire topic of general relativity impinges upon it, as documented in the references commented out. Brews ohare (talk) 18:56, 5 June 2009 (UTC)

Brews, that's the beginnings of a good section that you have written. But I do think that you need to take some note of what Rracecarr has said regarding clarifying the physical meaning to the lay reader.

I did Lagrangian many years ago, and I'm a bit rusty now. I never liked that topic. I don't ever remember doing centrifugal force in Lagrangian. The closest I came to it was with the grad(A.v) in the electromagnetic Lagrangian. Nevertheless, I have found what you have written to be very interesting.

It is clear to me that any velocity squared term will be a potential energy function for centrifugal force in hydrodynamics. What you need to do is somehow emphasize that point and link it up to the centrifugal potential energy of the parabolic curve in the water surface in the rotating bucket. You need to give the lay readers something physical to relate to. A few weeks back, I gave you a simplistic derivation of centrifugal force from transverse kinetic energy. See if you could get all that blended together. David Tombe (talk) 14:49, 7 June 2009 (UTC)

Are you suggesting a more detailed discussion of Hildebrand, for example? Brews ohare (talk) 15:14, 7 June 2009 (UTC)

Brews, I'm very rusty on Lagrangian and I'm not altogether sure what I am suggesting. But let me recall what I can remember about Lagrangian. It was based on conservation of energy, and generalized coordinates which were in turn based on the constraints and degrees of freedom that the geometry of a scenario presented.

So your generalized coordinates would normally be a variation on the ordinary vectors based on the constraints of the geometry.

I never did centrifugal force in Lagrangian explicitly, although I can now see that grad (A.v) in electromagnetism clearly points to the A.v term as being a centrifugal potential energy.

If I were you, I would tend to think up a practical example such as the rotating bucket of water. Then try and derive a Lagrangian based on the centrifugal potential energy.

I found what you wrote about Vi.Vi and Vi.Vj very interesting. It's true that only centrifugal force has an associated potential energy and that Coriolis force is a Vi.Vj situation. In my own books, both of them are v×ω. But in the case of centrifugal force, the v and the ω are inter-related such that we get Vi.Vi, whereas with Coriolis force, we have a constrained radial motion in a vortex and so the two v terms will be perpendicular and also, since it is a rigid vortex, ω gets multiplied by 2. I'll bet there will also be a Vi.Vk for an axial Coriolis force in gyroscopes. David Tombe (talk) 08:59, 8 June 2009 (UTC)

Wall of death

Above remarks by D. Tombe concerning Wall of death (motorcycle act): “For simplicity, I'll switch to a 'push' situation and consider a wall of death rider. The centrifugal force acts on the motorbike. The motorbike then transmits that effect to the wall, causing a pressure which induces a 'normal reaction'. The normal reaction causes a centripetal force. If the wall of death is circular, that centripetal force will be equal and opposite to the centrifugal force, but the two will not form an action-reaction pair. Nevertheless, Newton's concept of reactive centrifugal force is a faulty concept, based on his jealousy of Leibniz's equation, and so the centrifugal force pushing on the wall is indeed the so-called reactive centrifugal force as defined by Newton, even though in reality it is pro-active.

In the case of the wall of death, we can compare the centrifugal force to gravity in the sense that it acts on the motorbike, whereas we can compare the 'reactive centrifugal force' to 'weight', in that it is the knock-on pressure against a surface which induces the normal reaction.”

David: To me it is clear in this example that you have adopted the viewpoint of the motorcycle rider, that is, a local frame of reference. Everything you say is accurate from the viewpoint of that frame. See Centrifugal_force_(rotating_reference_frame)#Skywriter. However, it is not an inertial frame of reference. In the inertial frame, the centripetal force acting upon the bike is the net force found by vector addition of gravity and the normal force from the wall. This is the example of a banked turn discussed in Centrifugal_force_(rotating_reference_frame)#The_banked_turn.

As a general remark, intended as constructive, you place too little emphasis upon the effects of changing the reference frame. Brews ohare (talk) 16:17, 7 June 2009 (UTC)

Brews, reactive centrifugal force, polar coordinates, Leibniz's equation, and real rotating frames of reference, such as rotating platforms with dragging forces, or the rotating Earth with its entrained atmosphere, are all about inertial forces measured relative to the inertial frame.

Apparent deflections of stationary objects as viewed from a rotating frame of reference are something different that has got nothing to do with the inertial forces. David Tombe (talk) 09:21, 8 June 2009 (UTC)

This is an assertion on your part not in accord with the view I've stated above, and so far as I can tell, not supported within Newtonian mechanics. In the vector mechanics view of Newton, inertial forces do not exist in inertial frames (that is the definition of an inertial frame). In the Hamilton-Lagrange viewpoint, they do exist as "generalized" inertial forces, but this formulation cannot be taken as defining Newtonian forces. The generalized forces do not transform as vectors when frame changes are made, and indeed may involve multiple frames.

It appears to me that in attaching yourself to Leibniz, you put yourself directly in the historical path of Lagrange and Hamilton, that is, in a formulation that is based upon energy, not vectors, and upon the Euler-Lagrange equations, not Newton's laws. However, your intuitive interpretations are Newtonian and refer to vector forces, which is inconsistent with the Lagrangian formulation.

I have made this observation before, but you have not looked into it carefully. In terms of the planetary problem, a Lagrangian formulation in terms of ( r, θ ) is exactly your view, and involves no reference to rotating frames, as you frequently point out. In most cases, the Lagrangian viewpoint has at best an opaque connection to Newtonian forces, but in this particular example, the same equations stem from a Newtonian viewpoint in a co-rotating frame. This connection is confusing, particularly when the identity of the equations is taken to suggest the different conceptual backgrounds also are identical, a non-sequitor. "All roads lead to Rome", but that does not mean the "high road" and the "low road" are the same road. Brews ohare (talk) 13:58, 8 June 2009 (UTC)

You would find the discussion (available on line) by Edmond T Whittaker. A treatise on the analytical dynamics of particles and rigid bodies (Reprint of 1917 2nd ed.). Cambridge University Press. p. 40. extremely comfortable, including as it does exactly your view of centrifugal force as a potential. Brews ohare (talk) 17:21, 9 June 2009 (UTC)

Brews, I'll come back to Leibniz again, but let's first deal with Lagrangian. When I did Lagrangian mechanics, I couldn't see anything from a physics standpoint that differed in principle in any important regard from all the classical mechanics that I had done previously. But there may well have been differences from a mathematician's perspective such as you mention about the generalized coordinates not transforming as vectors.

Basically Lagrangian is based on the conservation of energy and analyzing a motion based on the constraints which the geometry dictates. So for example we might look at the motion of a ball inside a parabolic bowl. The degrees of freedom of motion and the constraints would be analyzed, a set of generalized coordinates would be chosen that could be used to locate the ball in terms of angles or distances, and an expression for both potential and kinetic energy would be expressed in terms of those generalized coordinates. That was what Lagrangian was all about. A Lagrangian was formed from the two energy terms, treated as a constant of the motion, and a solution was then found for the motion in terms of the generalized coordinates.

Lagrangian dealt with energy as energy is defined in relation to the conservative forces, which esentially means the radial forces, such as the radial centrifugal force where v^2 is involved. Lagrangian does not however adequately cater for the non-conservative forces such as the transverse Coriolis force or the axial Coriolis force where Vi.Vj and Vi.Vk vanish.

The centrifugal force in Lagrangian and the centrifugal potential energy is one and the same centrifugal force that appears as an inertial force in polar coordinates and in the Leibniz equation. It is measured relative to the inertial frame.

When you talk about Newtonian mechanics in relation to the centrifugal force, are you specifically talking about Newton's reactive centrifugal force? If so, you must remember that Newton's centrifugal force does not tie in with Newtonian mechanics. It is a specious concept which he thought up to denigrate Leibniz's equation. Newton was clearly jealous of Leibniz for completing the planetary orbital equation with the inverse cube law centrifugal force. Newton only got as far as the inverse square law force of gravity, whereas Leibniz got both. Newton's reactive centrifugal force was a nonsense idea based on a mis-application of his 3rd law of motion and it only ever had a modicum of sense when applied to the special case of circular motion. David Tombe (talk) 14:18, 10 June 2009 (UTC)

David, you say: "The centrifugal force in Lagrangian and the centrifugal potential energy is one and the same centrifugal force that appears as an inertial force in polar coordinates and in the Leibniz equation. It is measured relative to the inertial frame."

I would agree with the first sentence with some restrictions: (i) the statement applies only to the restricted example (ii) the term "centrifugal force" is used in a very particular sense of referring to a particular mathematical formula, and does not imply any general statement about the various conceptual frameworks that may be used to obtain the result. I would disagree 100% with the second statement: first, from a Newtonian vector mechanics viewpoint there is zero centrifugal potential energy in an inertial frame, and second, from a Lagrangian viewpoint no frame is specified, only the coordinates, so reference to an inertial frame in the Lagrangian context is unwarranted here. If you do take the time to read Edmond T Whittaker. A treatise on the analytical dynamics of particles and rigid bodies (Reprint of 1917 2nd ed.). Cambridge University Press. p. 40., the Lagrangian viewpoint will be clear. Brews ohare (talk) 15:04, 10 June 2009 (UTC)

Brews, I think that what you will need to do next is to give a practical scenario and analyze it using the Lagrangian method. You will need to find some scenario that involves absolute rotation and show what the generalized coordinates are in the context. David Tombe (talk) 12:32, 11 June 2009 (UTC)

See Edmond T Whittaker. A treatise on the analytical dynamics of particles and rigid bodies (Reprint of 1917 2nd ed.). Cambridge University Press. p. 40., Francis Begnaud Hildebrand (1992). Methods of Applied Mathematics (Reprint of 1965 2nd ed.). Courier Dover Publications. p. 156. ISBN 0486670023., V. B. Bhatia (1997). Classical Mechanics: With Introduction to Nonlinear Oscillations and Chaos. Alpha Science Int'l Ltd. p. 82. ISBN 8173191050.. What else is needed? Brews ohare (talk) 13:07, 11 June 2009 (UTC)

Brews, The reference is excellent but I meant a practical demonstration for the article itself. We need a practical demonstration for the lay reader. David Tombe (talk) 13:56, 11 June 2009 (UTC)

Polar Coordinates and the Leibniz Equation

Brews, when the Lagrangian section is completed with a practical demonstration, we will then need to move on to a section on polar coordinates and the Leibniz equation.

Meanwhile, can we get back to the general polar coordinate equation again and analyze the four terms for the inertial path (straight line) when the left hand side is equal to zero,

${\frac {d^{2}\mathbf {r} }{dt^{2}}}=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}$

and then compare these terms with the two terms in the Leibniz equation,

${\ddot {r}}=-k/r^{2}+l^{2}/r^{3}$

I need to know again what you think the two radial terms are in the polar coordinate equation. David Tombe (talk) 14:03, 11 June 2009 (UTC)

"Lagrangian Centrifugal force" = neoligism

The section entitled "Lagrangian Centrifugal Force" seems to be a novel synthesis, and the term itself is a neoligism as far as I can tell, since I've been unable to find any use of the term in any reputable source. It would be more accurate, and more reflective of actual sources, to refer to it as something like "generalized centrifugal force", or simply as the general definition of centrifugal force (in the inertial/fictitious sense). The current section in the article seems not to recognize that rotating rectilinear coordinate systems are just as "Lagrangian" as spatially curved coordinate systems. This is clearly explained in several reputable references that have been noted here previously, in which the fully general definition of fictitious/inertial forces is given, most with no reference at all to the Lagrangian formalism - although it is of course possible to express that definition (covering both rotating coordinates and spatially curved coordinates) in that formalism if one chooses to do so. Thus it's wrong to present the "first definition" as if it was not "Lagrangian", and then present the rest of the general definition as if it is uniquely "Lagrangian". The article should avoid novel synthesis, and should more closely reflect the actual referenced sources, and those sources should include the ones that have been provided that describe the general definition of centrifugal forces, which of course is not limited to rotating reference frames.Lagu2 (talk) 20:37, 13 June 2009 (UTC)

Lagu2, The concept of Lagrangian centrifugal force was new to me too. However, I felt that its inclusion by Brews assisted in breaking the logjam that had occurred due to other elements who were intent in suppressing references to all approaches other than the 'rotating frames of reference approach'. I found the section interesting, in that I did actually learn things from it, but I would agree that this information needs to be carefully grafted into something more general.

Yes, it needs to be expressed in more general terms, not linked to the Lagrangian formalism, which is just one way of expressing the subject, and certainly not the only way that the subject is expressed in the reputable literature.

By the way, when you say the Lagrangian approach was new to you "too", I'm not sure who else you are referring to. It wasn't new to me. My point was (and still is) simply that the term "Lagragian centrifugal force" is a neoligism, i.e., it doesn't appear in any reputable references, including the references that are cited in the section with that title. And my further point is that there's a good reason for why this term doesn't appear in reputable sources, the reason being that it conveys a false impression and a misunderstanding of the subject. ALL coordinate systems can be regarded as Lagrangian coordinates, although the term is usually reserved for coordinates that differ in some way from those in terms of which the true absolute accelerations correspond to second derivatives of the space coordinates with respect to the time coordinate. Now, among coordinates that would qualify as "Lagrangian" on this basis, we can (if we wish) make a further distinction, between those that are just temporally non-inertial and those that are spatially non-inertial, but this is purely a matter of convention and choice, and the literature contains many examples of both choices.

The overall problem here is that there is one group who are determined to suppress all references other than those which relate to rotating frames of reference, whereas Brews, who is open minded to alternative approaches, seems to think that every alternative approach represents a different kind of centrifugal force. I of course firmly believe that there is only one universal centrifugal force and that it is best taught in terms of either polar coordinates or the radial Leibniz equation (without any mention of the solar vortex).

While I recognize that there are many sources regarding the rotating frames of reference approach, I personally think that the rotating frames are superfluous as regards co-rotating situations unless there is a real rotating frame present such as the Earth, or a rotating platform. As regards situations in which there is no co-rotation, I do believe that the maths has gone off the rails as far as the physics is concerned. But that's my opinion. Let them put it into the article by all means. But that is no excuse to suppress a generalized centrifugal force section in which centrifugal force is explained without the involvement of rotating frames of reference.

In that respect, Brews's section is good because it gives one example of sourced material that deals with centrifugal force outside of the context of rotating frames. David Tombe (talk) 11:09, 14 June 2009 (UTC)

I agree with your last statement. Now if we can get similar coverage of the sourced material that presents the generally definition of centrifugal force (the definition that subsumes both the rotating frame and the "Lagrangian" and the modern spacetime approaches), it would be a big improvement to the article.Lagu2 (talk) 19:28, 14 June 2009 (UTC)

Yes, it's interesting, as it shows that if you treat r-double-dot as an acceleration, even though it's not one, then the equations you get for F=ma are the same as what you get in a co-rotating frame. It's another way to explain the CF term that looks like a force but is not one in reality. As far as David's claim that some are trying to suppress alternative sources or points of view, I don't think that's true. The only POV being suppressed is David's unique idiosyncratic one, in which he wants to redefine Coriolis forces and call most physics books incorrect in their treatment of centrifugal force. Dicklyon (talk) 18:13, 14 June 2009 (UTC)

Just to be clear, when you say "if you treat r-double-dot as an acceleration, even though it's not one...", hopefully we all understand that a similar statement applies for ALL definitions of the fictitious/inertial centrifugal force, including the rotating frame version. In every case, we are interpreting certain second derivatives of coordinates as if they were real accelerations, even though they are not. So, it's misleading to say that this "mis-identification" is unique to the "Lagrangian" treatment. This all just emphasises the importance of putting a single unified and general explanation of centrifugal force (in the fictitious/inertial sense) into the article. It's fine to distinguish between different sub-sets, such as temporal curved coordinates and spatially curved coordinates, but the article should present them as subsets of the single general definition, which is what they are. The reference posted by Tim Rias awhile back gives a good overview of the general modern definition. That reference should definitely be reflected in the article.Lagu2 (talk) 19:26, 14 June 2009 (UTC)

Right. Do that. Dicklyon (talk) 19:30, 14 June 2009 (UTC)

We're getting back to the same stalemate that arose on the Kepler's laws of planetary motion page. That stalemate entailed the inability of most editors to commit themselves on names for the terms. We have two equations which we need to get consensus on as far as names for the terms are concerned. We have the polar coordinates equation,

${\frac {d^{2}\mathbf {r} }{dt^{2}}}=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}$

and we have the Leibniz equation,

${\ddot {r}}=-k/r^{2}+l^{2}/r^{3}$

I would like to see what names are being suggested for the terms in the polar coordinate equation when it is equated to zero, as in the inertial path, and them compare the situation for the Leibniz equation. There is a twist to all of this which nobody seems to be prepared to address. If we could discuss this in a civilized manner, we may then be ready to write a generalized article on centrifugal force with a list of sections which deal with the various approaches such as,

(1) Rotating Frames/fictitious forces (2) Newton's reactive centrifugal force (3) Leibniz (4) Lagrangian (5) Polar coordinates (6) Special relativity (7) General Relativity (8) Quantum Mechanics

or even Maxwell and the young John Bernoulli with their seas of tiny vortices as opposed to Leibniz's large neo-Cartesian vortex, although that could be elaborated on in the history section.

Once we get over the intransigence that up until now has forbidden all but the rotating frames approach and the reactive approach to appear in the article, then I can see an end in sight to this long running edit war. David Tombe (talk) 07:23, 15 June 2009 (UTC)

The second equation is an immediate consequence of the first (for an inverse-square central force). The term l^2/r^3 in the second equation equals the term rw^2 in the first. This term is sometimes called a centrifugal force. It is an inertial force, also known as a fictitious force. It also corresponds to a Christoffel symbol, and it is sometimes simply treated as a component of the second derivative of the position vector. This should be covered in the discussion of fictitious/inertial definition of centrifugal force, based on the available reference sources that have been cited previously.

In your list of topics, items (1), (4), (5), and probably (6) are all under the heading of "inertial/fictitious forces". Your item (3) is unclear. Perhaps you have in mind a discussion of the Leibniz/Berkeley/Mach view of inertia? Your item (7) might refer to the Lens/Thirring effect, I suppose, or the more general issue of boundary conditions. Not sure what you have in mind for (8).Lagu2 (talk) 18:11, 15 June 2009 (UTC)

Lagu2, The second equation involves a gravitational field, whereas the first equation relates to an idealized situation which never exists in practice. As you know, when a small object is considered relative to a nearby larger object, the gravitational tail of the small object will protrude backwards from the small object in the direction away from both objects.

In the first equation, there are no gravitational tails involved. When we equate the first equation to zero, we will have an idealized straight line inertial path in which there is no gravitational tail on the windward side of the motion. In both the radial case and the transverse case, the two terms will cancel out mathematically. In the transverse case, this leads to the familiar conservation of angular momentum. In the radial case, it implies that there is an inertial centripetal force which takes the form $r{\dot {\theta }}^{2}$. I can only therefore assume that in the first equation, the centrifugal force is actually being represented by the ${\ddot {r}}$ term, and not by the $r{\dot {\theta }}^{2}$ term.

This may explain some of the confusion which arises due to the fact that some sources, including Goldstein and Taylor, refer to the $r{\dot {\theta }}^{2}$ term as the centripetal term in connection with polar coordinates, but yet they refer to that same term as the centrifugal term in connection with Leibniz's radial equation.

Some get around this dilemma by saying that the centripetal term becomes the centrifugal term when we move it to the other side of the equation. But that is clearly not a good enough explanation. An inward force cannot become an outward force simply by changing terms to the other side of an equation.

The truth is that the two equations represent different physical scenarios. The second equation is the radial planetary orbital equation which involves gravitational field tails. The first equation is an idealization which never exists in practice. That is why I always favoured putting the planetary orbital equation into the main article rather than a general treatment of polar coordinates. I always said to Brews, FyzixFighter, and Steve Byrnes, that the polar coordinates equation is no good until we create a real physical scenario in which to relate the terms to. David Tombe (talk) 11:09, 19 June 2009 (UTC)

Lagrangian centrifugal force

The term "Lagrangian centrifugal force" was introduced simply to keep the structure of the article parallel for all topics. This term is a convenient shorthand to refer to the cumbersome construction that involves explaining what a generalized coordinate is and then identifying the quadratic ${\dot {q_{k}}}$ terms as "centrifugal". This usage is explained clearly in the article.

Neoligisms are not appropriate for Wikipedia. Whenever you find it necessary to introduce a neoligism to express your point, you are creating a novel synthesis, which does not belong in Wikipedia (regardless of what you think its merits are, or how sure you are that you are correct.)Lagu2 (talk) 13:40, 17 June 2009 (UTC)

Lagu2: Frankly, I don't care a twit about this: using a specific wording in the title of a section and nowhere else is not a neologism (that is, it is not "coining a word"), it is simply a section title. Brews ohare (talk) 15:00, 17 June 2009 (UTC)

In fact, what is called "centrifugal force" in the Lagrangian formulation, in general, has very little connection with the Newtonian concept. As explained very carefully in the article, the centrifugal force in the Lagrangian formulation is not a Newtonian force, does not transform like a vector, and agrees only in very simple examples with the centrifugal force in a Newtonian formulation; and even then only when the appropriate observation frame and the appropriate coordinate system is chosen.

Centrifugal force (in the inertial/fictitious sense) is not a Newtonian force, regardless of whether we are speaking in terms of a rotating frame or spatially curved coordinates, so your point here is mistaken. Take a look at some of the references that have been provided, that explain the issue quite clearly (like the reference provided by Tim Rias not long ago).Lagu2 (talk) 13:40, 17 June 2009 (UTC)

Lagu2: Centrifugal force is a Newtonian force in a non-inertial frame. In an inertial frame it is zero, regardless of whether we use spatially curved coordinates. For example, it is so defined and used in:

John R Taylor (2005). Classical Mechanics. University Science Books. p. 342. ISBN 1-891389-22-X.

Vladimir Igorević Arnolʹd (1989). Mathematical Methods of Classical Mechanics (2nd ed.). Springer. p. 130. ISBN 978-0-387-96890-2.

Cornelius Lanczos (1986). The Variational Principles of Mechanics (Reprint of Fourth Edition of 1970 ed.). Dover Publications. p. 103. ISBN 0-486-65067-7.

LD Landau and LM Lifshitz (1976). Mechanics (Third ed.). Oxford: Butterworth-Heinemann. p. 128. ISBN 978-0-7506-2896-9.

Louis N. Hand, Janet D. Finch (1998). Analytical Mechanics. Cambridge University Press. p. 267. ISBN 0521575729.

and in the discussions (with supporting sources) in Centrifugal force (rotating reference frame) and inertial frame of reference. Brews ohare (talk) 15:00, 17 June 2009 (UTC)

By no stretch of imagination can the Lagrangian formulation be considered to be a "generalization" of the Newtonian vector force version; neither is it a "more general" formulation; the proof of this is that the Lagrangian formulation does not reduce to the Newtonian formulation except in a few special cases; in other, more general cases, both can be defined and are incompatible. As one example: Newtonian centrifugal force always is zero in an inertial frame of reference; the Lagrangian version can be (and often is) non-zero in an inertial frame. Brews ohare (talk) 04:18, 17 June 2009 (UTC)

Again, you're mistaken about this. Of course, one can define highly abstract Lagrangian coordinates, far removed from anything that would be regarded as space and time coordinates, but in general any coordination of a physical system (including inertial coordinates) can be classified as Lagrangian coordinates. It is simply a formalism, not a physical theory. And it is beside the point. As has been explained repeated and in great detail, the fictitious/ inertial forces arising in terms of a rotating frame are just a special case of the more general fictitious/forces that arise whenever a system of reference is used in which the spatial coordinates of inertial paths are not linear functions of the time coordinate. Again, I urge you to read some of the references that have been provided.Lagu2 (talk) 13:40, 17 June 2009 (UTC)

Lagu2: The words "in general" in general refer to the general case. In general is not "in specific", it's "in general". In general, in the Lagrangian formulation (according to sources cited repeatedly in the article) one finds the centrifugal force using the definition in terms of ${\dot {q_{k}}}^{2}$, with no restrictions upon the "highly abstract Lagrangian coordinates, far removed from anything". That is not "beside the point", that is the literature of several subject areas.

Lagu2: On your second set of points, you are confusing two disparate conceptions here: "the fictitious/ inertial forces arising in terms of a rotating frame" are absolutely not "just a special case of the more general fictitious/forces that arise whenever a system of reference is used in which the spatial coordinates of inertial paths are not linear functions of the time coordinate." The former are used to define inertial frames and are zero in such frames; the latter are defined regardless of the frame, vary with the coordinate system regardless of the frame, and may well be non-zero in inertial frames. The latter also can be expressed in terms of Christoffel symbols, which have bearing neither upon Newtonian vector mechanics nor mechanics in special relativity.

You may have in mind a context based upon general relativity, rather than either Newtonian vector mechanics or special relativity. I'd submit (i) that extension is beyond the scope of this article, and (ii) that extension would serve primarily to make things much more complex. Brews ohare (talk) 15:00, 17 June 2009 (UTC)

Reference Material for Brews

Here is the relevant quote from the reference provided by Tim Riaz a couple of weeks ago. Hopefully Brews can read this, and acknowledge that this reference states the point of view that there are usages of the term "centrifugal force" in the inertial/fictitious sense that are distinct from, and more general than, the rotating-frame definition, and that this reference (along with the several other that have been provided) has not be fabricated or falsified, and that it is from a reputable source, and that, therefore, this view ought to be represented accurately in the article (and that it does not mention the word "Lagrangian").

"This is not the whole story about centrifugal force in classical mechanics, since it makes its appearance in at least two other familiar contexts. As discussed in detail by Abramowicz,5 one is the train, plane, car context in which one has an accelerated platform to which a local reference frame is attached, essentially the previous problem with additional motion of the origin of coordinates. Any point fixed in this local platform will then experience accelerations tangential and transverse to its direction of motion, and the transverse acceleration can be interpreted in terms of a centrifugal force in the local reference frame due to its instantaneous rotation about the center of the osculating circle associated with the curvature of its path.

A third context in which the centrifugal force is usually introduced is in the discussion of motion in a central potential.5 Here one introduces a polar coordinate system in the plane of the motion and then expresses the equation of motion in that coordinate system. The radial component of this equation for general motion then contains what is interpreted as a centrifugal force term due to the curvature of the circular angular coordinate lines. This term is just the sign-reversal of the centripetal acceleration for motion confined to these coordinate lines and enters the equation of motion through a Christoffel symbol term associated with this curvature, quadratic in the angular speed... there is no rotating frame in this discussion..."Lagu2 (talk) 16:44, 17 June 2009 (UTC)

That's an interesting interpretation he has there; all three of those are what I would describe as rotating frames of reference, but this may be a way to better integrate the one-dimensional radial distance that David prefers to not describe as a rotating frame; it's still a fictitious force due to the fact that the one dimension co-rotates, of course. We'll also have to look closely at what he means by "the sign-reversal of the centripetal acceleration for motion confined to these coordinate lines"; the centripetal force on a planet in non-circular orbit, by the definition we agreed on, is not along the radial vector, but toward the center of the osculating circle, and the recriprocal-r-cube centrifugal force is not exactly the negative of the central force. Can you look more closely and figure out what he means exactly? Dicklyon (talk) 16:53, 17 June 2009 (UTC)

Can you repeat the ref you're referring to? Preferably with a link? Dicklyon (talk) 16:54, 17 June 2009 (UTC)

Here's a link to an online version [7] (see pages 4-6). And here's the last bit of the quote that Lagu2 ellipsised out:

"Although there is no rotating frame in this discussion, one may be introduced by letting the new system rotate about the center of force so that the particle in motion has a fixed new angular coordinate. In this rotating frame, the same centrifugal force term is then realized as in the rigid body discussion as a rotating frame effect, but rotating about the center of force, not the instantaneous radial direction associated with the curvature of the particle path. For circular motion, this centrifugal force is just the sign-reversal of the centripetal acceleration of the particle path, but for general noncircular motion, the two quantities are not simply linked.

In other words, the “fictitious” centrifugal force is a convenience that only has meaning with respect to some implied reference frame..."

--FyzixFighter (talk) 18:03, 17 June 2009 (UTC)

I fail to see how any reference to centrifugal force in an accelerating frame aids the argument that centrifugal force is being generalized beyond the Newtonian vector mechanics viewpoint. All such problems can be handled with fictitious forces that vanish in an inertial frame. Any departure from this viewpoint to support the notion that curved spatial coordinates lead to fictitious forces, on the other hand, inevitably lead to the Lagrangian context, whether explicitly recognized or not, and can be readily presented as Lagrangian formulations that are incompatible with the Newtonian vector mechanics. Brews ohare (talk) 17:53, 17 June 2009 (UTC)

But the referenced article does not interpret polar coordinates as generalized coordinates in the Largangian sense. Instead it interprets polar coordinates as defining what Bini calls a "nonlinear reference frame" (the later being a coordinate independent notion).

And before you disregard the reference as being in the context of GR not Newtonian mechanics, this is not exactly true. The context of the article is recovering Newtonian(like) mechanics as a limit of GR. Although the main body of the article is about extra subtleties introduced when trying to this in a space with actual curvature, the introduction quoted here however explains how some of the introduced notions apply to the normal Newtonian limit of GR, flat space. In particular, recovering Newtonian mechanics in this situation involves some arbitrary choices, namely the choice of frame. Different choices of frame lead to slightly different versions of Newtonian Mechanics. The canonical Newtonian choice is a frame which is specifically adapted to the flatness of spacetime, a so called linear or inertial frames (basically this uses the exponential map to extend the local frame at one point in spacetime to the whole of space time). You can however also make different (nonlinear) choices for the frame, each of which will lead to the introduction of new inertial/fictious forces. One is the familiar rotating one, in which one chooses a vorticity free congruence on the spacial slices, that rotates about a point as you move through time. You can however also choose a congruence that is not vorticity free, but specially adapted to the axialsymmetry of spacetime, but remains stationary over time, this is the nonlinear frame defined by polar coordinates.

This nicely unifies the different contexts in which there appears a fictious force labelled centrifugal. (TimothyRias (talk) 09:01, 18 June 2009 (UTC))

In terms of the very particular case of the central force problem, this relationship to Lagrangian mechanics has been argued ad nauseum and has been referenced to reliable sources (Hildebrand, Goldstein, and Bahtia) showing that generalized coordinates (r θ) in a Lagrangian viewpoint lead to the same results as a co-rotating frame in Newtonian mechanics, but without actually explicitly referring to a rotating frame. Isn't all this discussion no more than a re-hash of what is already well established? Brews ohare (talk) 17:58, 17 June 2009 (UTC)

Indeed it is. It has already been well established that the more general definition of the term centrifugal force in the context of classical mechanics is broader and more comprehensive that you (Brews) claim. Please understand that it's pointless for you to argue that all physical situations can be analyzed in terms of inertial coordinates with no fictitious forces, or in terms of a rotating coordinate system with the ratating-frame definition of fictitious forces. Of course they can. The point is that they can also be analyzed in terms of a larger class of coordinate systems, and quantities appear in those analyses that are called (in reputable sources) "centrifugal force", and this includes certain systems of coordinates that are not rotating. FredFyzic's inclusion of the elipses in the above quote is succumbing to the same logical fallacy of thinking that because A exists, B must not exist. This simply does not follow. You guys need to step back, take a deep breath, and think about this situation clearly. There is simply no doubt about the fact that reputable references state that terms which they call "centrifugal force" arise in terms of non-rotating coordinate systems. Anyone who denies this, in view of the provided references, is simply ...well.. mistaken.Lagu2 (talk) 18:26, 17 June 2009 (UTC)

Well, you do hold a lofty view, but so far have no basis for it beyond your assertions. Where are the citations and quotations? The contrary view is VERY well documented from reputable sources, not only page references but extensive word-by-word quotations. Brews ohare (talk) 19:26, 17 June 2009 (UTC)

Here is a link [8] (see pages 5-6), and here is the relevant quotation.

"This is not the whole story about centrifugal force in classical mechanics, since it makes its appearance in at least two other familiar contexts. As discussed in detail by Abramowicz,5 one is the train, plane, car context in which one has an accelerated platform to which a local reference frame is attached, essentially the previous problem with additional motion of the origin of coordinates. Any point fixed in this local platform will then experience accelerations tangential and transverse to its direction of motion, and the transverse acceleration can be interpreted in terms of a centrifugal force in the local reference frame due to its instantaneous rotation about the center of the osculating circle associated with the curvature of its path.

A third context in which the centrifugal force is usually introduced is in the discussion of motion in a central potential.5 Here one introduces a polar coordinate system in the plane of the motion and then expresses the equation of motion in that coordinate system. The radial component of this equation for general motion then contains what is interpreted as a centrifugal force term due to the curvature of the circular angular coordinate lines. This term is just the sign-reversal of the centripetal acceleration for motion confined to these coordinate lines and enters the equation of motion through a Christoffel symbol term associated with this curvature, quadratic in the angular speed... there is no rotating frame in this discussion..."Lagu2 (talk) 16:44, 17 June 2009 (UTC)

The same point is made in several over reputable reference texts that have been provided to you previously.Lagu2 (talk) 21:18, 17 June 2009 (UTC)

The link refers to General Relativity, which I do not feel is within the scope of the article as it presently exists, and would require considerable introductory material to include. The "contexts" described by Timothy that you quote verbatim are not germane for reasons presented and are not published sources, of course. Reference to "reference texts that have been provided to you" is a bit vague: perhaps you could provide them again? Brews ohare (talk) 21:37, 17 June 2009 (UTC)

The relevant quote begins "This is not the whole story about centrifugal force in classical mechanics", not in general relativity. Also, the authors of that preprint are recognized scholars in the field, and what they say in this introductory section (quoting other published sources such as Abromavic) is perfectly consistent with what has been said in numerous other published sources that have been provided to you repeatedly. Look, it's perfectly obvious that you are simply determined to continue spewing out utterly specious objections to every reference that says something other than your own personal point of view.Lagu2 (talk) 22:05, 17 June 2009 (UTC)

I misunderstood your earlier remark and thought you were attributing this lead to Timothy. However, the quote apparently is an excerpt from the linked pdf file (The intrinsic derivative and centrifugal forces in general relativity: I. Theoretical foundations) that refers to "as discussed in detail by Abramowicz" in an article not available to me, and only sketched in the linked pdf. The purpose of the cited article is framed in terms of general relativity. The discussion of the central force problem you quote from this pdf introduces the Christoffel symbols, just as I have discussed above, and points out exactly what I have said: "Although there is no rotating frame in this discussion, one may be introduced by letting the system rotate.... In this rotating frame, the same centrifugal force term then is realized as in the rigid body discussion as a rotating frame effect....." All this is perfectly in keeping with the Hildebrand, or Goldstein, or Bhatia references I cited above.

You are not reading what has been said and citing arguments that do not support your standpoint at all, but repeat what already has been discussed in detail and in context. If you wish to proceed, instead of name-calling ("Look, it's perfectly obvious that you are simply determined to continue spewing out utterly specious objections"), deal with the presented case and indicate why you think the cited and quoted work is at variance with the arguments already presented in the article proper. It isn't. Brews ohare (talk) 23:30, 17 June 2009 (UTC)

Your references and citations and arguments are all irrelevant to the point of this discussion, which is to determine whether, within the reputable literature, the term "centrifugal force" is sometimes used to refer to terms that arise in non-rotating coordinate systems due to the curvature of the spatial coordinates. It has already been established beyond any question that it is. The current article does not reflect this aspect of the concept of centrifugal force, so it should be revised to reflect this fact. It doesn't really matter how many irrelevant arguments you make, the only way you can legitimately challenge the matter is by showing that the cited reference which have been given to you several times now (e.g. Stommel and Moore) have been falsified. Since they have not been falsified, you really have no legitimate basis for your position.Lagu2 (talk) 01:55, 18 June 2009 (UTC)

I'm rather lost by this argument. Isn't the section that Brews added on the Lagrangian approach covering what Lagu2 is asking for already? What is the argument here? Dicklyon (talk) 03:15, 18 June 2009 (UTC)

Dick is entirely right about this: everything you say in the above paragraph, User:Lagu2, already is stated in the article proper with citations and complete discussion. More detail can be found here. Brews ohare (talk) 09:15, 18 June 2009 (UTC)

I think Lagu2's problem here (and I tend to agree with him on this) is that the article muffles away the centrifugal force as it appears in polar coordinates as a special case of the Lagrangian sense of the term, while we have a notable source that interprets in terms of (nonrotating) reference frames that has nothing to do with the Lagrangian use of the term. (for example it that case it will transform like a vector under local rotations. If anything the Lagrangian use of the term "centrifugal force" for any generalized fictitious force dependent on the square of the generalized velocity, is inspired on the fact this is the traditional nomenclature for the term of that form as it appears in the Lagrangian analyses of the central potential problem in polar coordinates. (TimothyRias (talk) 09:53, 18 June 2009 (UTC))

Maybe you can make that more clear by adding text to that effect, with link to source, so I can look at it in more specific terms. Dicklyon (talk) 14:28, 18 June 2009 (UTC)

I am not clear what the "notable source" is that Timothy refers to. I also find it unlikely that the source conflicts at all with the Newtonian view if in fact its centrifugal force transforms like a Newtonian vector force under coordinate changes. Finally, Timothy's view of the origin of the centrifugal identification of the ${\dot {q_{k}}}^{2}$ terms is perhaps historically interesting, but does not detract from the fact that this terminology is in active use with no such context provided. It's departure from the Newtonian view is very clear. The widespread usage of this terminology is why this approach is included in the article.

To that I'd add that the centrifugal force in polar coordinates is not "muffled away", but simply shown to be equivalent to a Newtonian vector mechanics analysis in a co-rotating frame. Brews ohare (talk) 21:08, 18 June 2009 (UTC)

Article by Abramowicz

This article, quoted by a secondary source provided by Lagu2, can be found at Centrifugal force - a few surprises From the abstract: "It is shown that in general relativity (and contrary to the situation in Newtonian theory) rotation of a reference frame is a necessary but not sufficient condition for the centrifugal force to appear. A sufficient condition for its appearance in the instantaneously corotating reference frame of a particle is that the particle motion in space (observed in the global rest frame) differs from a photon trajectory."

That this is a paper in general relativity, and that it introduces a co-rotating frame already does not bode well for its use as a supporting document for Lagu2. The introductory section refers to black holes, frame dragging and various and sundry other matters that are quite clearly outside the framework of the present Wiki article.

Abramowicz does, however, include a summary of Newton's ideas, entirely in accord with the present article. He makes the observation: "Since Newtonian dynamics can be fully discussed using inertial frames, one might argue that there is no need for the concept of centrifugal force. Indeed, Landau & Lifshitz quotes the term 'centrifugal force' only once." He goes on to say that nonetheless the concept is useful and proceeds to introduce the co-rotating frame. I have found nothing in this paper to support Lagu2. I did find his discussion would benefit D Tombe, especially where he says: "It seems that we always intuitively analyze dynamical effects of rotation in the corotating reference frames." Brews ohare (talk) 00:41, 18 June 2009 (UTC)