From: foltinek@math.utexas.edu (Kevin Foltinek) Subject: KAM theory (was Re: Do you recognise this book...) Date: 07 Jan 2000 12:11:41 -0600 Newsgroups: sci.math Summary: [missing] In article foltinek@math.utexas.edu (Kevin Foltinek) writes: > That would probably be KAM theory, for Kolmogorov, Arnol'd, Moser. Someone sent email (which I thought was also a post, but it appears to have not made it after two days) to me asking what KAM theory is. I don't know details, but my vague understanding is this. Given a completely integrable Hamiltonian system (M,\omega,h), where M is a 2n-dimensional manifold with symplectic form \omega and Hamiltonian function h, one can find "action-angle" coordinates (q,p). q is the "angle", p is the "action" (for reasons which will be made obvious below). Of course these are symplectic coordinates, so \omega = \sum_{i=1}^n dq^i \wedge dp_i . The dynamics of the Hamiltonian system are given by p = constant q = q0+t*v [on the torus], v=constant . If M is compact, these coordinates induce (or arise from, I'm not sure which) a local foliation of M by n-tori. In the non-compact case, the "torus" might be something like R^n, or R^{n-k} times a k-torus. (This foliation always happens on any symplectic manifold, but the special thing in this case is that it's somehow "compatible" with the additional structure obtained from the Hamiltonian function.) There's probably some local/global issue in all of this, but I'm not sure what exactly happens. Then you can perturb the Hamiltonian, h' = h + \epsilon f, and ask what happens to the perturbed oribts of the system. KAM theory addresses this question, and I think involves something about generic tori being stable, but I don't know KAM theory and never did so that's the best I can do. It's useful because it gives information about systems which are not completely integrable. Kevin.