Mechanical systems with their configuration space being a Lie group

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 $Hcolon,T^*G to mathbb{R}$ is $G$-invariant with respect to the natural action of $G$ on $T^*G$?