Classical mechanics motivation for poisson manifolds? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T16:47:11Z http://mathoverflow.net/feeds/question/28532 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28532/classical-mechanics-motivation-for-poisson-manifolds Classical mechanics motivation for poisson manifolds? Jan Weidner 2010-06-17T16:34:22Z 2010-06-18T05:02:37Z <p>Suppose I want to understand classical mechanics. </p> <p>Why should I be interested in arbitrary poisson manifolds and not just in symplectic ones?</p> <p>What are examples of systems best described by non symplectic poisson manifolds?</p> http://mathoverflow.net/questions/28532/classical-mechanics-motivation-for-poisson-manifolds/28539#28539 Answer by Aasmund Ervik for Classical mechanics motivation for poisson manifolds? Aasmund Ervik 2010-06-17T17:38:48Z 2010-06-17T17:38:48Z <p>Well, first of all, you would have to dive pretty far into classical mechanics in order to require either of these two tools. A typical introduction to the subject, based e.g. on the book of Goldstein, Poole and Safko, would not need such advanced mathematics.</p> <p>That being said, non-symplectic Poisson manifolds do occur in classical mechanics, when studying the time evolution of systems, most commonly in the transition from classical mechanics to quantum mechanics. </p> <p>A fundamental tool in quantum mechanics is the commutator between two operators, written e.g. [x,p]. The classical analogue of this is the Poisson bracket between two functions of the coordinates q_i, p_i and t, e.g. { f(q,p,t) , g(q,p,t) }.</p> <p>Now for the geometry: suppose that your coordinates q_i lie in a Riemannian manifold, Q. Then Q together with the two-form w, which is the exterior derivative of the so-called canonical one-form on the phase space, usually form a symplectic manifold.</p> <p>Now, a theorem (which I can't remember the name of) states that if (Q, w) is a symplectic manifold, then ( Q , {-,-} ) forms a Poisson manifold. This, again, turns out to be the requirement for constructing a Lie algebra homomorphism from the set of bounded sequences on Q to the set of vector fields on Q, and now we are very close to quantum mechanics.</p> <p>For a specific system: consider standard 3D Euclidean space. This is odd dimension, so not symplectic. Construct the Poisson bracket such: {x,y} = z , {z,x} = y, {y,z} = x, where x,y,z, are the coordinate functions, i.e. this is the cross product in awkward notation. Now let F be in C^inf(Q), consider {F,-} acting as a derivation on the set of all polynomials. Then, by Weierstrass theorem, you can approximate a definition of {F,G} for all F, G in C^inf(Q).</p> <p>So we have just constructed a very simple system (a point in Euclidean space) and represented it with a Poisson manifold. Note, however, that it has the set of all spheres centered at the origin as a foliation, and this is a foliation consisting of symplectic manifolds, which turns out to be a general feature.</p> http://mathoverflow.net/questions/28532/classical-mechanics-motivation-for-poisson-manifolds/28562#28562 Answer by Victor Protsak for Classical mechanics motivation for poisson manifolds? Victor Protsak 2010-06-17T19:58:37Z 2010-06-17T19:58:37Z <p>For many reasons and purposes, it is the Poisson bracket, not the symplectic form, that plays a primary role.</p> <ul> <li> Equations of motion and, more generally, the evolution of observables have an easy form: $$\frac{\partial f}{\partial t}=\{H,f\}.$$ <li> Conserved quantities form a Poisson subalgebra: $$\{H,F\}=\{H,G\}=0 \implies \{H,\{F,G\}\}=0.$$ <li> Since symmetries (Hamiltonian group actions) are Poisson by nature, the moment map is defined in the Poisson setting: $$M\ni P\mapsto (X \mapsto H_X(P)).$$ Unlike in the symplectic case, <em>both</em> steps in the Hamiltonian reduction, restriction to the level set and factorization by the action of the stabilizer, are naturally carried out in the Poisson category, even for singular reduction. <li> Quasiclassical approximation in quantum mechanics (and conversely, quantization of classical mechanical systems) is expressed via the Poisson bracket: $$[\hat{F},\hat{G}]=ih\widehat{\{F,G\}} +O(h^2).$$ </ul> <p>On the other hand, many natural systems have a degenerate Poisson bracket and/or are infinite-dimensional.</p> <ul> <li> The phase space of various tops is the dual space $\mathfrak{g}^*$ of a Lie algebra. This is a universal example of a linear Poisson structure. Symplectic leaves are the coadjoint orbits and the Poisson center is given by the (classical) "Casimir functions". <li> Classical integrable systems such as KdV admit a bi-Hamiltonian structure (i.e. a pair of compatible Poisson brackets). This has no analogues in symplectic theory. <li> Some of these structures are obtained by a reduction from a linear Poisson structure on a suitable infinite-dimensional Lie algebra (loop algebra, algebra of matrix differential or pseudo-differential operators, etc). </ul> <p>Finally, a related practical consideration: even if you are interested in studying a symplectic manifold, frequently this is best accomplished by embedding it as a symplectic leaf into a simpler Poisson manifold, <em>whether or not it has direct physical meaning.</em> In particular, this applies to flag manifolds of semisimple Lie groups, which are topologically complicated objects, but can be identified with coadjoint orbits, thus embedded into a vector space with a linear Poisson structure.</p> http://mathoverflow.net/questions/28532/classical-mechanics-motivation-for-poisson-manifolds/28599#28599 Answer by David Bar Moshe for Classical mechanics motivation for poisson manifolds? David Bar Moshe 2010-06-18T05:02:37Z 2010-06-18T05:02:37Z <p>The following <a href="http://philsci-archive.pitt.edu/archive/00002373/01/ButterfieldNHSympRed.pdf" rel="nofollow">article</a> by J. Butterfield suggests 3 reasons motivating Poisson manifolds (over symplectic manifolds)(page 81)</p> <ol> <li><p>Parameters and stability. Sometimes it is easier to analyze stability of the dynamics on a Poisson manifold than on its symplectic leaves.</p></li> <li><p>Naturality: The rigid rotator problem is more natural to analyze on SO(3) than on its coadjoint orbit.</p></li> </ol> <p>3.Reduction: When the configuration space is a Lie group, it is natural to use T*G/G as the reduced phase space. This is a Poisson manifold isomorphic to the dual of the Lie algebra and having a Lie Poisson structure.</p>