User:Jamontaldi/Relative equilibrium

A relative equilibrium is a type of motion studied in Mathematics and Physics, and to a lesser extent molecular chemistry. In an equilibrium there is no motion at all, while in a relative equilibrium there is motion but, roughly speaking, the shape of the system does not change. Relative equilibria are sometimes called stationary motions, describing for example the uniform rotational motion of a star in astrophysics or the pure rotation of a molecule.

The term relative equilibrium was introduced by Henri Poincaré (1885) in his study of the motion of a fluid mass, and given a general mathematical context by Stephen Smale (1970). The idea was already apparent in earlier work by Bernhard Riemann (1861), Edward Routh (1877), Jacobi (183?) and others.

Definition

Suppose a mechanical system (or more general dynamical system) has a group of symmetries G, with Lie algebra ${\mathfrak {g}}$. Then a relative equilibrium is a motion which coincides with the action of a 1-parameter subgroup of G. In more detail, let $x(t)$ be a solution to the dynamical system on a manifold M. Then this is a relative equilibrium if there is a $\xi \in {\mathfrak {g}}$ for which ${\dot {x}}(0)=\xi _{M}(x(0))$. Here $\xi _{M}$ is the vector field on M generated by $\xi$.

There are a number of useful equivalent conditions for a relative equilibrium. One is that a trajectory (solution) of the dynamical system is a relative equilibrium if it is contained entirely within a group orbit. Another is that the group orbit is invariant under the dynamics.

Examples

• A molecule undergoing a purely rotational motion with no vibrations; the symmetry group here would be the group SO(3) of rotations in space
• In celestial mechanics and the N-body problem, the well-known Lagrange configuration of 3 bodies lying at the vertices of an equilateral triangle, rotating uniformly about their centre of mass; the symmetry group here would be SO(2), the group of rotations in the plane about the centre of mass.
• A rigid body rotating about one of its principal axes; the group is the group SO(3) of rotations in space
• For the Riemann ellipsoid system, see Riemann (1861), or affine rigid body, the relative equilibria are precisely the motions classified by Riemann: those where the shape of the ellipsoid remains unchanged; the group here is the special linear group SL(3) of volume-preserving linear transformations in 3 dimensions.
• FLUIDS?

Bibliography

• Abraham, Ralph; Marsden, Jerrold E. (1978). Foundations of Mechanics (2 ed.). Addison-Wesley.

• Riemann, Bernhard (1861). "Ein Beitrag zu den Untersuchungen über die Bewegung eines flussigen gleichartigen Ellipsoides". Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen. 9: 3–36.

• Routh, Edward J. (1877). Treatise on the Stability of a Given State of Motion. MacMillan. Reprinted in 'Stability of Motion' (ed. A.T.Fuller) London 1975 (Taylor & Francis).

• Marsden, Jerrold E. (1992). Lectures on Mechanics (PDF). London Mathematical Soc. Lecture Notes. Vol. 174. Cambridge University Press. doi:10.1017/CBO9780511624001.

• Smale, Stephen (1970). "Topology and Mechanics II". Inventiones Math. 11: 45–64. doi:10.1007/BF01389805.

External links

• www.example.com