Periodic Orbits of Hamiltonian Systems, the Almost Existence Theorem, and Poisson Topology

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

Abstract Award: DMS-0307484 Principal Investigator: Viktor Ginzburg The present proposal focuses on two projects closely related to the principal investigator's previous work funded by NSF grants. These projects are the problem of existence of periodic orbits for Hamiltonian dynamical systems and the study of topological properties of certain Poisson manifolds. The first problem Viktor Ginzburg addresses in this proposal is the investigation of the size of the set of regular energy values on which a Hamiltonian system does not have periodic orbits. By the almost existence theorem, this set must be of zero measure and from counterexamples to the Hamiltonian Seifert conjecture it is known that this set may be non-empty. Thus the question is to bridge the gap between these two results. Another series of problems discussed in the proposal concerns the existence of periodic orbits for Hamiltonian systems of a special nature, including those describing the motion of a charge in a (strong) magnetic field or, more generally, the existence of periodic orbits near Morse-Bott non-degenerate symplectic extrema. These problems are closely related to the investigation of the (relative) Hofer-Zehnder capacity function and the Floer homology of certain Hamiltonians. The objective of the proposed research in the area of Poisson topology is to study connections between the geometry of Poisson structures and topology of underlying manifolds. Hamiltonian dynamical systems describe many classes of physical processes in which dissipative forces 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 and systems without such orbits have only been recently discovered. Yet, in all but simplest problems, finding periodic orbits requires the use of advanced and powerful mathematical methods. The investigation of periodic orbits lies at the very core of the modern theory of Hamiltonian dynamical systems. One of the main themes of the proposal is determining how large the collection of periodic/aperiodic energy values can be and showing that systems of a particular type carry periodic orbits of all energies. This class of systems includes those describing the motion of a charge in a magnetic field and the proposed research has potential applications to physics and mathematical aspects of mechanics. The last part of the proposal concerns the investigation of connections between geometrical and topological properties of a certain class of spaces arising in the study of systems with symmetries and in quantum mechanics.