Principal Investigator: Basak Gurel
This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The main long-term objective of the proposed work is to examine the reason for the existence of infinitely many periodic orbits for a variety of Hamiltonian dynamical systems. The first part of the proposal is comprised of several interconnected projects addressing this question for certain classes of Hamiltonian diffeomorphisms and also for specific Hamiltonian systems such as twisted geodesic flows. The PI will tackle these problems by making use of methods from symplectic topology including Floer homological techniques, spectral invariants, Hamiltonian Ljusternik-Schnirelman theory, the Seidel representation as well as methods from differential geometry such as h-principles. Among more specific tools pertinent to this investigation are local Floer homology, the mean index, the properties of action and index spectra and of action and index gaps, and geometric and dynamical properties of the symplectically degenerate maxima. The techniques utilized by the PI also have applications beyond establishing the existence of infinitely many periodic orbits. In particular, the PI's approach to the projects in the second part of the proposal draws heavily on one of her recent works on the periodic orbits problem. The PI outlines a method of proving that the diameter of the group of Hamiltonian diffeomorphisms is infinite for many symplectic manifolds, including some new instances. Another group of problems considered in this proposal concerns the symplectic topology of coisotropic submanifolds and some of its applications.
Hamiltonian systems constitute a broad class of physical systems where dissipative forces can be disregarded. For example, the planetary motion in celestial mechanics and the motion of a charged particle in a magnetic field are usually treated as Hamiltonian systems. One general, but not universal, feature of such systems is that they tend to have numerous periodic orbits. Corresponding to the 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. For a broad class of Hamiltonian systems, the number of periodic orbits is known to be infinite and this is thought to be the case for many (but not all) Hamiltonian systems. Yet, establishing the existence of periodic orbits often requires advanced and powerful mathematical tools. The proposal focuses on the existence problem for infinitely many periodic orbits of Hamiltonian dynamical systems in a variety of settings and on applications of the techniques used by the PI to attack this problem to some other related questions. The projects in the last part of the proposal concern a certain class of spaces which arise, for instance, in the study of Hamiltonian systems with symmetries. The proposed work is related to and has potential applications in mathematical physics, and geometric and quantum mechanics.
