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

• A "quasi-rigid" body in relativistic mechanics has an SO(3,1) configuration space, but the additional degree of freedom is a "gauge" one. The momenta (relativistic spin) then have the commutation relations of the generators of the Lorentz group. – Void May 20 at 14:08
• Fair enough, it's a different group, but on the level of physics it's still conceptually the same. – marmistrz May 21 at 11:16
• I do agree, I think there are very few classical, finite-dimensional systems where this applies. Even more, I think the most important applications will be in quantum field theories (EFTs). I guess that its just that you build the foundations starting from finite-dimensional classical systems. – Void May 21 at 11:39