Modern theory of separable hamiltonian systems

Abstract

Abstract In this thesis we present a modern, geometric (invariant) approach to the theory of autonomous (i.e. not explicitly depending on time) Hamiltonian integrable systems that are separable in the sense of Hamilton-Jacobi theory. We start with an elementary exposition of theory of Poisson manifolds, since Hamiltonian systems are dynamical systems defined on Poisson manifolds, and for that purpose we need to introduce some basic definitions from differential geometry, such as smooth (real) manifolds and tensor fields on smooth manifolds. In the next part of the thesis we introduce the reader to some basic facts about Hamiltonian systems and then about completely integrable (in the sense of Liouville) systems, presenting the famous Liouville-Arnold theorem describing the geometry of completely integrable systems and the existence of the so called action-angle variables that linearize the flow of any completely integrable system in a neighbourhood of any invariamt tori. We explain the basic ideas of Hamilton-Jacobi theory of finding solutions to a Hamiltonian system by looking for additively separable solutions of the related Hamilton-Jacobi equation. We put this theory in the modern language of separation relations on Poisson manifolds. Next we focus on a very important class of separable systems that are called St¨ackel systems, that is separable systems generated by separation relations that are linear in Hamiltonians and quadratic in momenta (and on their subclass generated by a single hyperelliptic separation (spectral) curve). Of special interest to us are St¨ackel systems of Benenti type that is St¨ackel systems with the St¨ackel matrix in the form of a Vandermonde matrix. For St¨ackel systems of Benenti type and generated by a single separation curve, we investigate the problem of finding canonical maps that turn the Hamiltonians of these systems into polynomials. We present two such maps: to the so called Vi`ete coordinates (which was previously known) and to what we call in this thesis Newton coordinates, i.e. coordinates generated by sums of powers of separation coordinates. The second possibility was discovered only very recently and in this thesis we present our own, alternative proof of Buchstaber and Mikhailov result; we also analyze in detail all the features of Benenti Hamiltonians in Newton coordinates, such as their iv Killing tensors, their special conformal Killing tensor and their pseudoriemannian metric. These results are new. We finish the thesis by pointing out some interesting directions for the future research. The thesis is richly furnished with examples. Keywords and phrases: Hamiltonian systems, integrable systems, HamiltonJacobi theory, sepration relations, separation curves, St¨ackel separable systems, Benenti class, AMS 2010 Subject Classification: 70H20, 53D17, 37K10, 70H06, 70G45