nLab
Hamiltonian dynamics on Lie groups

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 formall integrable systems.

Details

Let $G$ be a Lie group. Write $𝔤$ for its Lie algebra.

Choose a Riemannian metric

$$⟨-,-⟩\in {\mathrm{Sym}}_{C^{\mathrm{\infty }}(G)}^2\Gamma (TG)$$\langle -,-\rangle \in Sym^2_{C^\infty(G)} \Gamma(T G)

on $G$ which is left invariant?.

On the tangent bundle $TG$ this induces the Hamiltonian

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

This is now also called the Euler-Arnold equation.

Examples

• For $G=\mathrm{SO}(n)$ the special orthogonal group the above yields is rigid body dynamics. The left invariant metric $⟨-,-⟩$ 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.

Related concepts

• Euler-Arnold equation

References

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

• Ralph Abraham, Jerrold Marsden, Foundations of Mechanics