nLab
Hamiltonian dynamics on Lie groups

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

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 formall integrable systems.

Details

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

Choose a Riemannian metric

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

on GG which is left invariant?.

On the tangent bundle TGT G this induces the Hamiltonian

H:(vTG)12v,v. H : (v \in T G) \mapsto \frac{1}{2}\langle v,v\rangle \,.

This is now also called the Euler-Arnold equation.

Examples

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

Created on August 29, 2011 22:37:23 by Urs Schreiber (131.211.238.66)