I'm a looking for examples of dynamical systems that have multiply-connected compact configuration spaces.
Since I'm not a 100% sure about the correct terminology for the systems (I am sure about the the kind of space I want). What I'm looking for are systems in which energy is conserved and whose equations of motion can be determined (at least in principle) using classical Lagrangian mechanics.
The only ones I know of are simple pendula. E.g. the configuration space of a single pendulum is the circle $S^1$, the configuration space of a double pendulum is the 2-torus $\mathbb{T}^2$ etc.
Another possible example might be identical particles in a box. The configuration space of $n$ identical particles moving in open space is $\mathbb{R}^{3n}/S_n$, the orbit space of the group action on $\mathbb{R}^{3n}$ by the group of permutations of $n$ objects. I'm not sure whether limiting the particles to a box makes the configuration space compact though, and i certainly have no idea what its fundamental group would be.
I am cross posting this on physics SE. (https://physics.stackexchange.com/questions/265571/examples-of-multiply-connected-compact-configuration-spaces)