Hamiltonian Dynamical Systems and N-Body Problem

University Of Cincinnati Main Campus, Cincinnati OH

Investigators

Abstract

DMS-0071494 Dr. 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-dimensional algebraic sets which vary with the masses, energy and angular momentum of the particles. These computations should refute Birkhoff conjecture, yield information on the existence of cross sections, and answer the Birkhoff question on cross sections in the negative as they did in the non-regularized problem. As another application of the rich topology found in the integral manifolds of the three-body problem, he will use variational arguments to establish new classes of periodic solutions. Dr. Meyer will also work on (i) simplifying the proof of the existence of Xia diffusion in the three-body problem; (ii) the existence of comet-like periodic orbits in the N-body problem; (iii) establishing the stability of an equilibrium point of a Hamiltonian system in the case of 1:1 resonance; and (iv) the evolution and bifurcation of invariant manifolds of a Hamiltonian system which depend on a parameter. Dr. Meyer will work on a number of problems in the qualitative theory of the equations which describe the motion of mechanical systems and celestial bodies, namely Hamiltonian systems. One of the important equations of this type is the N-body problem which describes the motion of N masses -- the planets. Systems of this type typically conserve energy and momentum and these conservation laws place global restrictions on the possible motion of the bodies. Dr. Meyer has an ongoing research program studying the ramifications of these conserved quantities on the nature of the motion. Dr. Meyer will also investigate the existence of regular and chaotic motions in these systems and how these types of motions depend of various physical parameters. One type of regular motion he will establish is periodic motions in the N-body problem and one type of chaotic motion he will try to establish is known as Xia diffusion. Dr. Meyer will also study the evolution of the stable manifold of equilibrium solutions i.e. the evolution of the set of solutions which tend to an equilibrium (the stable manifold).