Hamiltonian dynamics on Lie groups



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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.


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.



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 (