What are some conserved quantities of Poisson brackets? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T16:16:15Z http://mathoverflow.net/feeds/question/1470 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1470/what-are-some-conserved-quantities-of-poisson-brackets What are some conserved quantities of Poisson brackets? Hung Tran 2009-10-20T18:54:43Z 2010-02-16T15:04:11Z <p>Poisson brackets play the very important roles in Symplectic geometry and Dynamical system. I'm interested in some conserved quantities of Poisson brackets.</p> <p>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. </p> <p>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.</p> <p>The question is, in general setting or in some particular case, can you find some conserved quantities S such that {H,S}=0?</p> <p>One easy and important example is for S= f(H) for f is smooth then we have {H,f(H)}=0.</p> <p>But can you have some other conserved quantities S?</p> <p><strong>Edit:</strong> 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.</p> <p>Any deeper intuitions or physical examples?</p> http://mathoverflow.net/questions/1470/what-are-some-conserved-quantities-of-poisson-brackets/1520#1520 Answer by Ari for What are some conserved quantities of Poisson brackets? Ari 2009-10-20T23:07:56Z 2009-10-20T23:07:56Z <p>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.</p> http://mathoverflow.net/questions/1470/what-are-some-conserved-quantities-of-poisson-brackets/1575#1575 Answer by Mike Usher for What are some conserved quantities of Poisson brackets? Mike Usher 2009-10-21T03:32:25Z 2009-10-21T03:32:25Z <p>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 <a href="http://en.wikipedia.org/wiki/Hamiltonian%5Fvector%5Ffield" rel="nofollow">Hamiltonian vector fields</a> span a k-dimensional <i>isotropic</i> 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. </p> <p>If there is an independent set of n functions, including H, which mutually Poisson commute, then the Hamiltonian system associated to H is called <a href="http://en.wikipedia.org/wiki/Integrable%5Fsystem" rel="nofollow">integrable</a>, and can be solved exactly using action-angle coordinates--see Arnold's book <i>Mathematical Methods of Classical Mechanics</i>. The example you gave is integrable, but in general this is a fairly-rarely satisfied condition.</p> <p>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. </p> <p>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 T<sup>n</sup>xR<sup>n</sup>. </p> <p>One positive statement that can be made is the following. If you look at the proof of <a href="http://en.wikipedia.org/wiki/Darboux%27s%5Ftheorem" rel="nofollow">Darboux's theorem</a> 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<sub>i</sub>,K<sub>i</sub>) around in terms of which the symplectic form is given by \sum dH<sub>i</sub>dK<sub>i</sub> <i>and H<sub>1</sub>=H.</i> So (H=H<sub>1</sub>,H<sub>2</sub>,...,H<sub>n</sub>) 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.</p> http://mathoverflow.net/questions/1470/what-are-some-conserved-quantities-of-poisson-brackets/1639#1639 Answer by Hung Tran for What are some conserved quantities of Poisson brackets? Hung Tran 2009-10-21T12:14:21Z 2009-10-21T12:14:21Z <p>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.</p> <p>@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</p> <p>"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."</p> <p>So, one practical and useful question may be:</p> <p>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.</p> <p>@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.</p>