Hamiltonian Dynamical Systems and N-Body Problem

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DMS-0071494Dr. Meyer will work on a variety of problems involving the $N$-body problem,Hamiltonian systems and dynamical systems. Building on his recent work with Dr. McCord on the integral manifolds of the spatial three body problem he will study the geometry and homology of the {\it regularized} (compactified)integral manifolds of the spatial three-body problem. These are eight-dimensionalalgebraicsets which vary with the masses, energy and angular momentum of the particles. These computations should refute Birkhoff conjecture, yield information on theexistence of cross sections, and answer the Birkhoff question on cross sections in the negative asthey did in the non-regularized problem. As another application of the richtopology found in the integral manifolds of the three-body problem, he will usevariational arguments to establish new classes of periodic solutions. Dr. Meyerwill also work on (i) simplifying the proof of the existence of Xia diffusionin the three-body problem; (ii) the existence of comet-like periodic orbits in the N-body problem; (iii) establishing the stability of anequilibrium point of a Hamiltonian system in the case of 1:1 resonance; and(iv) the evolution and bifurcation of invariant manifolds of a Hamiltoniansystem which depend on a parameter. Dr. Meyer will work on a number of problems in the qualitative theory of theequations which describe the motion of mechanical systems and celestial bodies, namelyHamiltonian systems. One of the important equations of this type is the N-bodyproblem which describes the motion of N masses -- the planets. Systems of this type typically conserveenergy and momentum and these conservation laws place global restrictions onthe possible motion of the bodies. Dr. Meyer has an ongoing research programstudying the ramifications of these conserved quantities on the nature of the motion. Dr. Meyer will alsoinvestigate the existence of regular and chaotic motions in these systems andhow these types of motions depend of various physical parameters. One type ofregular motion he will establish is periodic motions in the N-body problem and one type of chaotic motion he willtry to establish is known as Xia diffusion. Dr. Meyer will also study theevolution of the stable manifold of equilibrium solutions i.e. the evolution ofthe set of solutions which tend to an equilibrium (the stable manifold).