# Examples of multiply-connected compact configuration spaces

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. By configuration space I mean the space of points that each represent a different state of the system, e.g. the circle for a pendulum, each point on the circle corresponds to an angle between the pendulum and the vertical.

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 math SE. (https://math.stackexchange.com/questions/1845285/examples-of-multiply-connected-compact-configuration-spaces?noredirect=1#comment3774853_1845285)

• I'm fairly sure that the $\mathbb{R}^{kn}/S_n$ space you describe is simply connected. For simplicity, consider two identical particles in a 2D box. Given two loops in the space, we want to contract it into a small loop where each particle just moves a bit and returns to its starting place. If the two particles return to their starting locations without ever meeting, then contract each path individually, and you're done. If the two particles exchange positions, then bring their two paths together to a common point; have them "trade off" and go back to their starting locations. If they (cont) – Alex Meiburg Jun 30 '16 at 21:21
• meet in the middle somewhere, just arbitrarily break them apart in some way and proceed as above. This strategy clearly works in any number of dimensions, and I think the process can be repeated on each pair of particles to do so for > 2 particles in a box. That being said, this idea suggests another example: A particle confined to exist within a torus has a configuration space of a torus, which is multiply-connected. I wonder what $\mathbb{T}^n/S_n$, the configuration space for $n$ identical particles in a torus, looks like? – Alex Meiburg Jun 30 '16 at 21:23
• What about $SO(3)$? Think up a configuration space for this! – ClassicStyle Jun 30 '16 at 21:24
• @Alex: I don't quite understand your explanaition. The reason I think $R^{3n}/S_n$ is multiply connected is that I think $R^{3n}$ is the universal cover for that space which would mean $\pi_1(R^{3n}/S_n)\cong S_n$. – Nikolai Jun 30 '16 at 21:43
• @AlexMeiburg You are supposed to assume that two particles can't have the same position. Then the configuration space is multiply connected. I guess its cheating to say that a point particle constrained to a multiply connected surface? – Brian Moths Jun 30 '16 at 22:06