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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 $T^*G$?

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  • $\begingroup$ Do things like robotic arms count as naturally occurring for you? $\endgroup$
    – Mark Grant
    May 22, 2021 at 12:01
  • $\begingroup$ Yes, but it has the configuration space $SO(3)^k$, so it's still essentially the same. $\endgroup$
    – marmistrz
    May 22, 2021 at 15:25
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    $\begingroup$ It's not just invariant Hamiltonian that are interesting in this context. You can also have collective Hamiltonians: if $\mu : T^* G \rightarrow \mathcal{G}^*$ is the momentum map and $h$ is a function in $\mathcal{G}^*$, you can consider the Hamiltonian $H := h \circ \mu$. This is used to model a few mechanical systems that are not quite as symmetric as those that are $G$-invariant, but still "smell" of symmetry. In that context I've seen other groups used. Check out the paper sciencedirect.com/science/article/abs/pii/0003491680901554 $\endgroup$
    – alvarezpaiva
    May 22, 2021 at 18:10
  • $\begingroup$ Well, there may also be planar revolute joints parameterized by $S^1$, but I take your point. $\endgroup$
    – Mark Grant
    May 22, 2021 at 18:11

3 Answers 3

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Many of the standard classical mechanics examples have continuous symmetries. Most are invariant under translations in time. Many are invariant under spatial translations and rotations, e.g., isolated central force problems such as planetary motion. These are the fairly obvious ones, but for example for the two-body Kepler problem, there is an additional not so obvious symmetry associated with the Runge-Lenz vector. Generalizing to relativistic mechanics, one has the larger Lorentz group. Of course, if you allow yourself to think in terms of phase space rather than configuration space, you have the entire machinery of canonical transformations at your disposal.

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  • $\begingroup$ Of course, symplectomorphisms of the phase space are a much larger class than the lifted symmetries of the configuration space. The question is about something different: demanding that the phase space is of form $T^*G$ for a Lie group $G$ and that moreover the Hamiltonian is $G$-invariant w.r.t. cotangent lift of the action by left multiplication is extremely restrictive. $\endgroup$
    – marmistrz
    May 22, 2021 at 15:30
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    $\begingroup$ @marmistrz - Ok - well, does the enhanced symmetry of the Kepler problem ($SO(4)$ for negative energies, $SO(3,1)$ for positive ones) fit your bill? Maybe not, since the extra symmetry isn't directly associated with a cyclic coordinate in the action ... $\endgroup$
    – Michael Engelhardt
    May 23, 2021 at 0:25
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    $\begingroup$ While it has the $SO(4)$ symmetry group, it doesn't have $T^*SO(4)$ as its phase space, its phase space $T^*\mathbb{R}^3 \simeq \mathbb{R}^6$ (I'm talking about the reduced mass variant) $\endgroup$
    – marmistrz
    May 23, 2021 at 10:42
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Other examples of Hamiltonian systems with phase space the cotangent bundle of a group are coming from lattice gauge theory. There, a configuration is a map that assigns to every edge of the lattice an element of the structure group $G$ (usually taken to be a compact Lie group), which corresponds to the parallel transport along that edge. Accordingly, the configuration space is identified with $G^{\# \text{edges}}$ and the phase space is $T^* G^{\# \text{edges}}$. Note however that the natural action of the symmetry group $G^{\# vertices}$ is by conjugation, so that the reduced phase spaces are not coadjoint orbits but rather complicated objects (singular in general). See, e.g, section 9.4 in Rudolph & Schmidt: "Differential Geometry and Mathematical Physics" for more details.

Apart from this, most examples are indeed infinite-dimensional.

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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 $SO(3)$.

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