TITLE: Mechanical systems with their configuration space being a Lie group
QUESTION [8 upvotes]: Cross-posted from Physics.SE
In Marsden, Ratiu - Introduction To Mechanics And Symmetry there is a certain focus on reducing cotangent bundles of Lie groups. More precisely, if $G$ is a Lie group, then $T^*G$ is naturally a symplectic manifold. Then,

Lie-Poisson reduction provides a Poisson structure on the Lie coalgebra $\mathfrak{g}^*$ and a reduced Hamiltonian $\mathfrak{g}^* \to \mathbb{R}$. In other words, the dynamics can be analyzed on the reduced space $\mathfrak{g}^*$.
symplectic reduction allows us to reduce the phase space further and identify the reduced phase space with a coadjoint orbit $\mathcal{O} \subset \mathfrak{g}^*$, where $\mathcal{O}$ is equipped with the KKS symplectic structure.

However, apart from the configuration space $G = SO(3)$ which is the configuration space for the rigid body, all the examples provided by Marsden and Ratiu are infinite-dimensional.  I failed to come up with any other non-trivial finite-dimensional example, i.e. $\mathbb{R}^n \times (SO(3))^k$ doesn't count.
Are there any other naturally occurring mechanical systems, whose configuration space is a Lie group $G$ and that the Hamiltonian $H\colon\,T^*G \to \mathbb{R}$ is $G$-invariant with respect to the natural action of $G$ on $T^*G$?

REPLY [2 votes]: If one dimensional $n$ body systems fit your definition of a "naturally occurring mechanical system", then the paper by M.A. Olshanetsky and A.M. Perelomov, "Completely Integrable Hamiltonian Systems Connected with Semisimple Lie Algebras", Inventiones mathematicae (1976) Volume: 37, page 93-108 provides examples apart from just $SO(3)$.