What are some mechanics examples with a globally non-generic symplecic structure?

Question

In the framework of statistical mechanics, in books and lectures when the fundamentals are stated, i.e. phase space, Hamiltons equation, the density etc., phase space seems usually be assumed to be $\mathbb R^{2n}$, where maybe the $q^i$-coordinates are cut off to get a finite volume.

In the books about Hamiltonian mechanics, especially mathematical books, one needs a symplectic space $(\mathcal{M},\omega)$ and of course the Hamiltonian. Now necessarily, locally $\omega$ looks like the canonical form $\Theta=\text dq^i\wedge\text dp_i$.

Are there some relevant classical mechanics problems where one can state a less trivial $\omega$, and that globally?

I would like to see a global expression which is different from $\Theta$ (and also not just $\Theta$ in different global coordinates). That would be a nontrivial form, which might maybe arise over a more topologically complicated space than $\mathbb R^{2n}$, maybe due to restrictions of a mechanical system. And maybe you get such a form after a phase space reduction, but I don't actually know any explicit mechanical problem you need it for.

Answer 1

Phase spaces which are not cotangent bundles can be realized in mechanical systems with phase space constraints . The phase space given by Arnold: the two sphere $S^2$ can be mechanically realized as the reduced dynamics of an energy hypersurface of a two dimensional isotropic harmonic oscillator:

$ |p_1^2|+|p_2^2|+|q_1^2|+|q_2^2| = E$

We observe that the Hamiltonian generates a constant rotation rate in the $(p,q)$ planes, namely:

$ (p_i(t)+iq_i(t)) = exp(-iE_it) (p_i(0)+iq_i(0)) $

Thus we may choose to look at the system from the point of view of a "rotating system in phase space" in which the vector in the $(p_1, q_1)$ plane is always in the direction of $q_1$. Of course, we cannot do that on both planes because we have only one degree of freedom. Thus we are left with:

$|p_2^2|+|q_1^2|+|q_2^2| = E$,

which is just the equation of a two-sphere. Thus the reduced dynamics of a constant energy hypersurface is on a two-sphere.

The symplectic form has to be proportional the area of the sphere, because it is the volume form of the sphere and a two sphere has only one volume form.

This approach gives us a very big bonus upon quantization. It is well known that from the quantization of a sphere we get spin quantization. From the point of view of the isotropic oscillator for $E = 2j \hbar$, ($j$ is half integral), this quantization corresponds to the following energies of the individual oscillators: $(2j, 0), (2j-1, 1), .,.,., (0,2j)$. As can be seen there are exactly (2j+1) states as in the spin system.

The full theory of quantizations allows to write the corresponding wave functions also in the coordinates of the two sphere. Thus, we actually quantized the isotropic oscillator using spin quantization.

The equivalence of this procedure to the standard quantization of the isotropic harmonic oscillator is a very celebrated theorem by Guillemin and Sternberg called "Quantization commutes with reduction". Actually, this is the principle we apply when we quantize gauge theories (although there is no formal proof for the infinite dimensional case). You can find on the net numerous works on this subject.

Answer 2

Generally, coadjoint orbits of a Lie group provide important examples of global symplectic manifolds. In general such systems are obtained by symplectic reduction from a more fundasmental description.

For example, the spinning top is modelled for constant $J^2$ on a symplectic manifold $S^2$ that is a coadjoint orbit of the rotation group $SO(3)$. It is obtained by symplectic reduction from the $N$-particle model of a rigid body. (If $J^2$ is not taken fixed, one needs a more general description in terms of a 3-dimensional Poisson manifold.)

There are lots of more advanced such models. See the book Mechanics and Symmetry by Marsden and Ratiu.

More generally, Hamiltonian dynamics in Poisson algebras is also not just a mathematical game but is important in applications. For example, the Hamiltonian description of realistic fluids needs an infinite dimensional Poisson manifold. For the Euler equation, see, e.g.,
P.J. Morrison, Hamiltonian description of the ideal fluid, Reviews of Modern Physics 70 (1998), 467.
http://www.ph.utexas.edu/~morrison/98RMP_morrison.pdf

Answer 3

Any two dimensional closed orientable surface can have structure of a symplectic manifold (you can set your symplectic form equal to volume form). Moreover it will be "nontrivial" in the sense of being different from cotangent bundle of some other manifold. Also once you are given with some symplectic manifold you can always define a classical mechanical system on it, by introducing a Hamiltonian function and writing corresponding time evolution equations.

One explicit example is torus which can be obtained from phase space $R^2$ of a single particle by making following identifications on position and momentum :

$x+L_1=x$

$p+L_2=p$

So now any function $H(x,p)$ which is periodic in $x$ and $p$ with periods $L_1$ and $L_2$ respectively can serve as a Hamiltonian function on torus.