Periodic Orbits of Hamiltonian Systems, Cobordisms and Geometric Quantization, and Poisson Geometry

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

DMS-0072202 Viktor L. Ginzburg The present proposal focuses on several long-term projects and continues principal investigator's previous work funded by an NSF grant. The first question addressed in the proposal is the Hamiltonian Seifert conjecture or, more specifically, the existence problem for Hamiltonian dynamical systems without periodic orbits on a sequence of regular energy levels. The Hamiltonian Seifert conjecture is closely related to the next group of questions considered in the proposal. These questions lie in the area of symplectic topology and concern the existence of periodic orbits for Hamiltonian systems describing the motion of a charge in a magnetic field. The second part of the proposal includes a series of problems in Poisson geometry. Among these problems are, for example, the existence questions for equivariant Poisson moment maps and the construction of Poisson traces corresponding to the leaves of the symplectic foliation. A general program relying on applications of equivariant cobordisms to the study of Hamiltonian actions of compact groups is the subject of the concluding part of the proposal. Hamiltonian dynamical systems describe many classes of physical processes in which dissipation of energy can be neglected. For example, planetary motion in celestial mechanics and some electro- or magneto-dynamical processes can be, and usually are, treated as Hamiltonian dynamical systems. One of the classical subjects in the theory of dynamical systems is the study of periodic orbits (i.e. cyclic motions). Periodic motion is the simplest and most common type of motion after equilibrium. It is believed that a vast majority of Hamiltonian systems have periodic orbits. The first problem addressed in the proposal is the construction of Hamiltonian systems without periodic orbits. This is a question of considerable importance for the theory of Hamiltonian dynamical systems because examples of such systems would further advance our understanding of Hamiltonian dynamics. The next problem concerns the existence of periodic orbits for Hamiltonian systems describing the motion of a charge in a magnetic field. This class of Hamiltonian systems naturally arises in applications in physics and mechanics. However, few of the extremely powerful general methods that have been recently developed in symplectic geometry are applicable to this class of systems. The investigation of these systems should extend the limits of existing methods and result in the development of novel ones. Other problems considered in the proposal concern the study of connections between geometrical properties of classical-mechanical systems and certain quantum-mechanical phenomena.