Mathematical Sciences: Dynamics of Hamiltonian Systems

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9401740 Mather Mather intends to continue his work on dynamics of Hamiltonian systems. In the past, much of his work has been on dynamics of area-preserving mappings and autonomous systems in two degrees of freedom. More recently, he has extended part of this work to more degrees of freedom, in his recent paper, "Variational Construction of Connecting Orbits." However, this paper is only a beginning in this direction, in the sense that it should be possible to obtain similar results under much more general hypotheses. Proving the existence of connecting orbits between minimal sets under more general hypotheses will be a major part of Mather's research. As an example of what one might do with such results, consider the restricted three-body problem in the plane, i.e. the mechanical problem with a Newtonian gravitational potential, two massive bodies and one body of zero mass, all in the plane. In the case that the two massive bodies move about one another in circular orbits, Moser showed, in the early 60's, that if the zero mass body moves in an approximately circular orbit a long distance from the massive bodies, then it does not escape to infinity. Thus one has stability. This is part of KAM theory. In contrast, if the massive bodies move about one another in elliptical orbits, no such result is known. Mather's methods might lead to a proof that such a result is false. There are, however, major hurdles to be overcome before this program can be carried out. ***