Talk:Principal axis theorem

Section on Motivation?

Wikipedia is not a textbook. I think that section needs to be cut severely.--345Kai (talk) 03:54, 7 September 2011 (UTC)

I would agree. It might be better to give a more general statement of the result first. then have sections discussing 2D exmaples, 3D examples, and broader generalisations. Jheald (talk) 15:36, 21 June 2013 (UTC)

Axis or axes ?

Should the theorem be the "principal axis theorem" or "principal axes theorem" ?

Google shows a moderate preference for the former, both in general search results, and also in Google book and Google scholar searches.

On the other hand, I was first introduced to it as the "principal axes theorem", in the context of the three principal axes (plural) of the moment of inertia; and it seems to me that what the theorem discusses and provides is not just one principal axis, but the whole set of principal axes, so the plural form seems to me the more appropriate name, better describing what the theorem actually concerns.

I wonder if the singular form has come about by comparison with the parallel axis theorem, which in contrast does indeed concern the moment of inertia about a single axis parallel to a body axis.

What do others think? Jheald (talk) 15:45, 21 June 2013 (UTC)

In many important cases, there is only one axis that is special. That is, the problem has spheroidal symmetry. This is a common case in birefringence leading to the extraordinary ray. Or the moment of inertial of an American football. I suspect that it is common enough that principal axis comes up often in web searches, though the more general case is covered here. Gah4 (talk) 18:48, 6 May 2020 (UTC)

History

A several-paragraph account of the history of theorem, through various stages of development, is given at the start of

• L. A. Steen, Highlights in the History of Spectral Theory, American Mathematical Monthly, Vol. 80, No. 4 (Apr., 1973) (pp. 359-381) JSTOR,

recently reprinted in Marlow Anderson et al (eds., 2009), Who Gave you the Epsilon?: And Other Tales of Mathematical History, Mathematical Association of America. ISBN 0883855690, p. 36 (Google Books) Jheald (talk) 15:54, 21 June 2013 (UTC)

discriminant

It is nice to have the full eigenvalue expansion explanation, but the way I learned this in high-school (before I had enough math to do eigenvlaues), we had the form $ax^{2}+bxy+cy^{2}+dx+ey+f=0,$ with the discriminant $b^{2}-4ac$, which tells you ellipse (or degenerate ellipse) vs. hyperbola, and, left out of this article, the parabola case when it is zero. Since this can quickly find the answer for the common cases, it might be reasonable to add. Gah4 (talk) 18:56, 6 May 2020 (UTC)

IMO, the discriminant does not belong to this article, which is about a theorem that does not implies the discriminant. In fact, a conic section, and, more generally, a quadric have two different discriminants: the discriminant of the quadric, and the discriminant of its homogeneous part of degree two. This is explained in the three first sections of Discriminant#Generalization. This should also be described in Quadric, but the word "discriminant" is not even written there, although, for a quadric surface, the sign of the discriminant is the sign of Gaussian curvature. In Conic section, only the discriminant of the homogeneous part of degree two is called a discriminant, although the other discriminant is also mentioned, without being named (it is zero if and only if the conic section degenerates into lines. A better coherency between these three articles would be useful, D.Lazard (talk) 20:11, 6 May 2020 (UTC)

Should there be principal axes, more general and not the theorem? This comes up in different places, especially related to physics. I have wondered where it should be described and linked. One that I think isn't so obvious to so many people is the meaning of cubic symmetry in crystals. In jewelry, cubic zirconia is well known, but maybe not so well known why. Natural zirconia is monoclinic, and so birefringent. That is, the index of refraction is different along different axes. Cubic crystals are not birefringent. They have the same index of refraction on any axis, parallel to the crystal axes or not. I was trying to find a page to link that explains that materials with three equal principal axes have no special axis. Their index of refraction or moment of inertia is the same on any axis (in the latter case, through the center of mass). Gah4 (talk) 21:52, 6 May 2020 (UTC)

As far as I know, this article is unrelated with crystallography. The mathematics that underlies crystallography are more related to lattices and their symmetries (crystallographic restriction theorem, and Bravais lattices). D.Lazard (talk) 03:14, 7 May 2020 (UTC)

Well, not crystallography itself, but optics of crystalline materials. Non-cubic crystals have different index of refraction along different axes, and with principal axes as described here. The result is birefringence. Cubic crystals don't have birefringence. Gah4 (talk) 03:46, 7 May 2020 (UTC)

In the sense of this article, a principal axis is related to a quadratic form, or equivalently to a symmetric matrix. I had a look to the theory presented in Birefringence. This convinced me that there are symmetric matrices there, but, without further study, I am unable to know which one is a constant of a given crystal, and thus to which matrix refer the principal axes you are talking of. By the way, the discriminant of a quadratic form is also called its Hessian determinant, and the Hessian matrix of a multivariable scalar-valued function is the matrix of a (differential) quadratic form attached to the function. These multiples terminologies may make difficult to navigate between related WP articles. D.Lazard (talk) 09:08, 7 May 2020 (UTC)

It is the Electric susceptibility tensor which is a scalar for cubic crystals, and a symmetric tensor, at least at optical frequencies, otherwise. It then sets the index ellipsoid such that the index of refraction is axis dependent. Gah4 (talk) 12:38, 7 May 2020 (UTC)

Science

Mathematics in unconventional ways to improve your spoken English skills and experience is not build 49.37.99.90 (talk) 11:54, 23 July 2022 (UTC)