CAREER: Periodic Orbits of Hamiltonian Diffeomorphisms and Reeb Flows

The University Of Central Florida Board Of Trustees, Orlando FL

Investigators

Abstract

Hamiltonian systems constitute a broad class of mechanical systems where energy dissipation can be neglected. For example, the planetary motion in celestial mechanics, the flow of an incompressible ideal fluid, and the motion of a charged particle in an electro-magnetic field are usually treated as Hamiltonian dynamical systems. One of the most important questions concerning the dynamics of such systems and connected to many other branches of mathematics and physics is the existence of periodic orbits. Corresponding to cyclic motion, this is the simplest dynamical phenomenon after equilibrium, and an investigation of periodic orbits of a system is crucial in understanding its global behavior. To give but a few applications, the knowledge of periodic orbits is crucial in astronomy, particle accelerators, and fluid dynamics or can be used to understand stability of solutions for large times. Hamiltonian systems tend to have numerous periodic orbits, but proving the existence of even one closed orbit often requires advanced and powerful mathematical tools. This research project aims at establishing the existence of infinitely many periodic orbits for a broad class of Hamiltonian systems and analyzing the systems that fall outside this class, and the work will advance our understanding of the dynamics of conservative systems and result in the development of new powerful techniques applicable to other questions. Many of the systems considered in the proposal (e.g., magnetic flows) are of interest in physics and engineering, and some of the projects are expected to have applications in mathematical physics, geometric mechanics, and other areas. The research program focuses on the question of the existence of infinitely many periodic orbits for Hamiltonian dynamical systems in a variety of settings and on new methods to investigate this question that are currently being developed by the PI. The research comprises several interconnected projects addressing various aspects of this question for certain classes of Hamiltonian diffeomorphisms and Reeb flows and also for specific Hamiltonian systems such as magnetic flows. The project also opens up new research directions such as the study of non-contractible periodic orbits on closed manifolds. The PI will tackle these problems employing symplectic topological methods including Floer and quantum homological techniques, contact and symplectic homology, J-holomorphic curves, and spectral invariants, and will continue to develop new Floer theoretic techniques tailored for the study of periodic orbits. The projects have applications to other questions of interest in symplectic and contact dynamics and topology, classical dynamical systems, and Riemannian geometry.