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

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AbstractAward: DMS-0307484Principal Investigator: Viktor GinzburgThe present proposal focuses on two projects closely related tothe principal investigator's previous work funded by NSFgrants. These projects are the problem of existence of periodicorbits for Hamiltonian dynamical systems and the study oftopological properties of certain Poisson manifolds. The firstproblem Viktor Ginzburg addresses in this proposal is theinvestigation of the size of the set of regular energy values onwhich a Hamiltonian system does not have periodic orbits. By thealmost existence theorem, this set must be of zero measure andfrom counterexamples to the Hamiltonian Seifert conjecture it isknown that this set may be non-empty. Thus the question is tobridge the gap between these two results. Another series ofproblems discussed in the proposal concerns the existence ofperiodic 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 periodicorbits near Morse-Bott non-degenerate symplectic extrema. Theseproblems are closely related to the investigation of the(relative) Hofer-Zehnder capacity function and the Floer homologyof certain Hamiltonians. The objective of the proposed researchin the area of Poisson topology is to study connections betweenthe geometry of Poisson structures and topology of underlyingmanifolds.Hamiltonian dynamical systems describe many classes of physicalprocesses in which dissipative forces can be neglected. Forexample, planetary motion in celestial mechanics and someelectro- or magneto-dynamical processes can be, and usually are,treated as Hamiltonian dynamical systems. One of the classicalsubjects in the theory of dynamical systems is the study ofperiodic orbits (i.e. cyclic motions). Periodic motion is thesimplest and most common type of motion after equilibrium. It isbelieved that a vast majority of Hamiltonian systems haveperiodic orbits and systems without such orbits have only beenrecently discovered. Yet, in all but simplest problems, findingperiodic orbits requires the use of advanced and powerfulmathematical methods. The investigation of periodic orbits liesat the very core of the modern theory of Hamiltonian dynamicalsystems. One of the main themes of the proposal is determininghow large the collection of periodic/aperiodic energy values canbe and showing that systems of a particular type carry periodicorbits of all energies. This class of systems includes thosedescribing the motion of a charge in a magnetic field and theproposed research has potential applications to physics andmathematical aspects of mechanics. The last part of the proposalconcerns the investigation of connections between geometrical andtopological properties of a certain class of spaces arising inthe study of systems with symmetries and in quantum mechanics.