Variational Methods in Hamiltonian Mechanics

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PI: John N. Mather, Princeton UniversityDMS-0245336AbstractThe aim of this project is to construct orbits of Hamiltonian systems by variational methods. The focus is on questions related to Arnold diffusion. For small perturbations of strictly convex integrable systems in 2 or 3 degrees of freedom, Mather has recently succeeded in proving a strong form of Arnold diffusion. For perturbations in a cusp residual set, there are orbits whose action varies in a fairly arbitrary prescribed way. A major goal of this project will be to generalize these results to small perturbations of strictly convex systems in 3 degrees of freedom.This project deals with fundamental mathematical questions that arise in various physical application, e.g. the containment of a plasma in a tokomak (for the production of energy through nuclear fusion) and the question of the stability of orbits in planetary systems. In each case, it has to do with whether certain orbits of dynamical systems wander. In the case of the tokomak, the mathematical questions is related to the physical question of whether the hydrogen atoms in the plasma collide with the walls of the container; in the case of the solar system it is related to the question of whether the planets remain orbiting the sun for all time, or whether the mutual gravitational attraction of the planets could cause one of the planets to wander away from the sun over a very long period of time. It needs to be pointed out, however, that this project deals with a fundamental mathematical problem that originated in the study of physical problems of this sort; it does not address these problems directly.