Mathematical Sciences: Existence and Multiplicity Questions for Periodic Solutions of Hamiltonian Systems and Related Topics

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The theme of this work centers on the structure of solutions of systems of ordinary differential equations. The equations have roots in classical mechanics described by Hamiltonian systems. To understand the orbit structures of these systems, one tries to find invariant subsystems, the simplest of those being periodic solutions. This leads naturally to boundary value problems. Three different questions will be treated in this project. The first is concerned with conditions one can expect to find periodic solutions on a prescribed energy surface. Complete results are available for convex and star-shaped surfaces. Work will be done in expanding known theory to compact hypersurfaces of contact type. The expected criterion for periodic solutions is the vanishing of the first (real) homology group. A newly observed phenomenon of almost existence of periodic solutions will also be pursued. A second direction this work will follow is one of finding symplectic invariants for periodic solutions on compact hypersurfaces in symplectic manifolds and deciding on multiplicity questions. In attempting to find the number of periodic solutions on an energy surface, one is faced with the difficulty that the usual variational approach does not give the unparametrized periodic solutions. For compact convex surfaces, efforts will be made to establish this number - which is thought to be at least half the dimension of the ambient space. The third area of concentration will deal with symplectic geometry. Considerable activity has been stimulated by the solution of one of Arnold's conjectures and the introduction of holomorphic maps of Gromov. Current work will focus on the role of the 2nd homotopy group in the Lagrangian intersection problem.