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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 formally integrable systems.
Let be a Lie group. Write for its Lie algebra.
Choose a Riemannian metric
on which is left invariant?.
On the tangent bundle this induces the Hamiltonian
This is now also called the Euler-Arnold equation.
For the special orthogonal group the above yields is rigid body dynamics. The left invariant metric is the moment of inertia of the body.
For the infinite-dimensional Lie group of volume-preserving diffeomorphism of a compact smooth manifold, the above yields is Euler’s equations of hydrodynamics.
The original influential article is
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.