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 $T^*G$?

  • 1
    $\begingroup$ 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. $\endgroup$ – Void May 20 at 14:08
  • $\begingroup$ Fair enough, it's a different group, but on the level of physics it's still conceptually the same. $\endgroup$ – marmistrz May 21 at 11:16
  • $\begingroup$ 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. $\endgroup$ – Void May 21 at 11:39

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