Mathematical Sciences: Hamiltonian Dynamical Systems

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Abstract Meyer Dr. Meyer (with Dr. McCord) will study the topology of the integral manifolds of the regularized spatial three-body problem. He wishes to describe the bifurcations by computing the homology of these manifolds for various values of the parameters. He will try to clarify the existence of Arnold diffusion in the three-body problem using reduction, symplectic scaling, normalization, and the theory of normally hyperbolic manifolds. Dr. Meyer (with Dr. Schmidt) will study the existence of KAM tori near elliptic orbits of the three-body problem. He (with Ms Howison, a graduate student) will establish the existence of various periodic solutions of the spatial restricted three-body problem. He will also try to establish the existence of elliptical comet like orbits in the full three-body problem. Dr. Meyer studies the differential equations the describe the motion of the planets and satellites in our solar system, the N-body problem. He also studies other types of equations that describe mechanical systems. Since these equations are not solvable exactly, he studies questions about special types of solutions. For example, when does the system of equations have periodic solutions or even quasi-periodic solutions? He will attempt to establish several new families of periodic solutions and several new families of quasi-periodic solutions for the N-body problem. He has developed several new methods to attack these problems. The existence of these quasi-periodic solutions give some partial understanding of the stability of the solar system. He will also study some global questions about these systems. What global constraints on the possible configuration of the bodies come from the conservation of energy and momentum?