nLab Hamiltonian dynamics on Lie groups




physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics



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 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 \,.

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



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.