How can we design the global structure of the phase space?

Question

I want to know how to design a classical mechanical system that has a phase space $M$ with a nontrivial global topology. If I naively consider a system in which the generalized coordinate $q_1,\cdots,q_n$ lives on some manifold $M_q$ and if I take the standard kinetic energy $K=\sum_{i=1}^n\frac{1}{2}\dot{q}_i^2$ then the corresponding momentum is $p_i=\dot{q}_i$, which takes the value on $\mathbb{R}$. So, in this case the phase space takes the form $M=M_q\times \mathbb{R}^n$. Is there any natural classical mechanical setting in which the phase space is a hypersphere or torus?

I guess the classical spin $\vec{S}$ with fixed length lives on a sphere $S^2$.

Answer 1

A simple recipe is given by Hamiltonian mechanics. You just need an even dimensional manifold with a symplectic structure (odd dimensional manifolds cannot have a symplectic structure). Then, just take a Hamiltonian on the manifold and you get a perfectly well defined dynamical system.

A natural setting of this is probably Euler's top. You could think that phase space is 6 dimensional being $T^*SO(3)$ (explicitly, you can parametrise $SO(3)$ with the three Euler angles and you have their time derivatives which gives the conjugate momenta). However, due to rotational invariance, the system reduces to an effective 2D phase space homeomorphic to a sphere. A pedestrian approach would be just to write Euler's equations of motion: $$ \dot L+\Omega\times L = 0 \\ \Omega = J^{-1}L $$ with $L$ the angular momentum, $J$ the momentum of inertia and $\Omega$ the angular velocity. The phase space is the range of possible $L$, which lies in a sphere since $|L|$ is conserved. The Hamiltonian is the usual kinetic energy: $$ H = \frac{1}{2}L\cdot J^{-1} L $$ and symplectic form is the usual area element of the sphere. By analogy with quantum mechanics, you can also model a gyromagnetic spin: $$ H = -\gamma B\cdot L \\ \dot L = \gamma L\times B $$ with $B$ the magnetic field and $\gamma$ the gyromagnetic ratio.

Note that when your system has a more complicated topology (i.e. is not simply connected), you can loosen the previous recipe by not requiring that $H$ is a globally defined function but a locally defined function. A pedestrian approach is to notice that the EoM's depend only on the derivatives of $H$, so if the different branches differ by a local constant, the EoM's are defined unambiguously even if $H$ is multivalued. More abstractly, you need a closed 1-form to construct a conservative vector field, which need not be globally exact, but can be made locally exact.

This can be useful for example when the phase space is a torus for example, which has two "holes" unlike the sphere. Qmechanics' answer already gives a nice construction of a physical example. Note that it is similar in spirit with the Euler top, since you start with a bigger phase space $T^*\mathbb R^2/\mathbb Z^2$ that you reduce thanks to symmetries. Another classic example is the standard map. Note that it is a discrete time conservative system.

Hope this helps.