nLab Hamiltonian dynamics on Lie groups

Contents

Idea

Many important special cases of classical mechanics involve physical systems whose configuration space is a Lie group, for instance rigid body dynamics but also (for infinite-dimensional Lie groups) fluid dynamics.

All these systems have special properties, notably they are formally integrable systems.

Let $G$ be a Lie group. Write $\mathfrak{g}$ for its Lie algebra.

\langle -,-\rangle \in Sym^2_{C^\infty(G)} \Gamma(T G)

on $G$ which is left invariant?.

On the tangent bundle $T G$ this induces the Hamiltonian

H : (v \in T G) \mapsto \frac{1}{2}\langle v,v\rangle \,.

The Euler-Arnold equation is the associated differential equation governing the velocity of geodesic flow.

For $G = SO(n)$ the special orthogonal group the above yields is rigid body dynamics. The left invariant metric $\langle -,-\rangle$ is the moment of inertia of the body.
For $G$ the infinite-dimensional Lie group of volume-preserving diffeomorphism of a compact smooth manifold, the above yields is Euler’s equations of hydrodynamics.

The original influential article is

Vladimir Arnol'd, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16 (1966) fasc. 1, 319–361. (MathSciNet)

A standard textbook reference is section 4.4 of

Last revised on October 28, 2022 at 18:15:55. See the history of this page for a list of all contributions to it.