Contents

# 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.

## Details

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

Choose a Riemannian metric

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

on $G$ which is left invariant?.

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

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

This is now also called the 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

Last revised on February 7, 2019 at 07:09:45. See the history of this page for a list of all contributions to it.