Periodic orbits of Hamiltonian systems and symplectic topology of coisotropic submanifolds

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

The present proposal focuses on two projects closely related to the PI's previous work. The first group of problems addressed concerns the so-called Conley conjecture and the almost existence theorem for periodic orbits. The general form of the Conley conjecture asserts the existence of infinitely many periodic points of a Hamiltonian diffeomorphism of a symplectically aspherical, closed manifold. This conjecture has recently been established by Hingston and the PI. However, many aspects of the problem require further investigation. For instance, one can expect the conjecture to hold even when the manifold is not aspherical, but the Hamiltonian diffeomorphism has sufficiently many fixed points. The proposed research addresses this and some other aspects of the Conley conjecture. The almost existence theorem guarantees the existence of periodic orbits on almost all levels of a proper, autonomous Hamiltonian for a broad class of symplectic manifolds. This theorem is closely related, on both conceptual and technical levels, to the Conley conjecture and the Weinstein conjecture. A project described in the proposal aims at complementing the almost existence theorem by showing that the set of energy values without periodic orbits is nowhere dense. The second part of the proposal focuses on symplectic topological properties of coisotropic submanifolds. These properties generalize the Lagrangian intersection property and the Maslov class rigidity and have important application in dynamics. Moreover, a general picture is emerging, enabling one to treat such facts as non-existence of exact Lagrangian embeddings and the existence of closed characteristics on a contact type hypersurface as particular cases of one phenomenon. The main goal of the program started recently by the PI and outlined in the proposal is to further analyze and extend this picture. 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 lying at the very core of modern theory of Hamiltonian dynamical systems and symplectic geometry is the study of periodic orbits (i.e., cyclic motions). Periodic orbits are ubiquitous: a vast majority of Hamiltonian systems have periodic orbits and the number of distinct periodic orbits is infinite for a broad class of systems. The analysis of this phenomenon, building on the PI's recent work, is among the main objectives of the proposed research. The class of dynamical systems in question 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.