What are some conserved quantities of Poisson brackets?

Question

Poisson brackets play the very important roles in Symplectic geometry and Dynamical system. I'm interested in some conserved quantities of Poisson brackets.

Let's say we are working on T^n x R^n (T^n is the torus in R^n, T^n = R^n/Z^n). Assume that I have the Hamiltonian H: T^n x R^n \mapsto R, where H=H(x,p) and H is smooth for simplicity.

Every function here is T^n-periodic in x. For K=K(x,p) then we define the Poisson bracket {H,K} = D_p H.D_x K - D_x H.D_p K.

The question is, in general setting or in some particular case, can you find some conserved quantities S such that {H,S}=0?

One easy and important example is for S= f(H) for f is smooth then we have {H,f(H)}=0.

But can you have some other conserved quantities S?

Edit: here's one specific example: If H(x,p)=H(p)+V(x_1+x_2+...+x_n) for V: R \mapsto R is T-periodic then this kind of Hamiltonian has some other conserved quantities S=p_i - p_j for any i \ne j.

Any deeper intuitions or physical examples?

Answer 1

My impression is that for most choices of H this is a hard question. In general, the largest possible cardinality of a set of independent Poisson commuting functions on a 2n-dimensional symplectic manifold (where by independent I mean that their differentials are linearly independent at each point in their domain--in particular H and f(H) are not independent) is n. Sketch proof: Given k independent Poisson commuting functions, their Hamiltonian vector fields span a k-dimensional isotropic subspace (i.e., the symplectic form dxdp vanishes on the subspace), and it's an easy linear algebra exercise to show that the largest possible dimension of an isotropic subspace is n.

If there is an independent set of n functions, including H, which mutually Poisson commute, then the Hamiltonian system associated to H is called integrable, and can be solved exactly using action-angle coordinates--see Arnold's book Mathematical Methods of Classical Mechanics. The example you gave is integrable, but in general this is a fairly-rarely satisfied condition.

There are some obstructions to H having independent functions that Poisson commute with it. For example, Poincare noticed that if the Hamiltonian vector field of H has a periodic orbit, so if the orbit has period T the time-T map F of the flow has some fixed point y, then looking at the derivative of F at y gives the following restriction: if there are k Poisson-commuting functions (including H) which are independent along the orbit through y, then the derivative of F at y has to have the eigenvalue 1 with multiplicity at least 2k. The generic situation is that 1 occurs as an eigenvalue with multiplicity just 2, so in this sense the typical Hamiltonian doesn't have any other functions that Poisson commute with it along its periodic orbits.

Another point that occurs to me is that if almost every level set of H contains a dense orbit for the Hamiltonian flow of H, then since any K such that {K,H}=0 is constant along the Hamiltonian flow of H it would follow that any such K would be constant on almost every level set of H. But then K would be constant on every level set of H just by continuity, and this is only possible if K has the form f(H) that you mentioned. However I'm not sure if the assumption that almost every level set of H has a dense orbit is realistic--in particular I can't think of any examples where this holds on TⁿxRⁿ.

One positive statement that can be made is the following. If you look at the proof of Darboux's theorem that's given in Arnold's above mentioned book, you'll see that it proves the following slightly stronger statement: if y is any point at which dH is nonvanishing, then there are local coordinates (H_i,K_i) around in terms of which the symplectic form is given by \sum dH_idK_i and H₁=H. So (H=H₁,H₂,...,H_n) is an n-tuple of Poisson commuting functions, just defined near the given point y. So locally there are certainly functions that Poisson commute with H--they just can't generally be extended globally.

Answer 2

Not sure if this is what you're asking for, but conserved quantities correspond to symmetries of the Hamiltonian (Noether's theorem). In the example you just gave, I think the conserved quantity p_i - p_j corresponds to the action sending x_i to x_i + y while sending x_j to x_j - y, which leaves the Hamiltonian invariant. In fact, you can take x_i to x_i + y_i for i=1,...,n, where y_1 + ... + y_n = 0, since this leaves the sum x_1 + ... +x_n (and hence the Hamiltonian) unchanged.

Answer 3

Since any conserved quantity K here is required to be T^n-periodic in x variable. So, we generally can't have some conserved quantities like x_i - x_j. T^n here is the simple case of compact manifold.

@Mike: The answer of Mike is really interesting. I thank Mike for his response. The things I have seen are quite close to your observations, in particular

"But then K would be constant on every level set of H just by continuity, and this is only possible if K has the form f(H) that you mentioned."

So, one practical and useful question may be:

Give one level set of H, i.e. {(x,p)| H(x,p)=c} for some c fixed provided that this level set is not empty. Can we find some conserved quantities K such that {H,K}=0 on this level set only. Even some examples other than f(H) or p_i - p_j in the special case will be really interesting.

@Ari: I see your point. But I don't think we can't point out any new conserved quantities other than p_i - p_j.