Mathematical Sciences: Dynamical Systems

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The two investigators are working on different projects. Each, however, is related to Kolmogorov - Arnold - Moser theory. The principal investigator, Mather, is working on a project to prove the existence of "Arnold diffusion" in small perturbations of integrable Hamiltonian systems of positive normal torsion. The hope is to solve this problem, or at least a version of it, by variational methods. Forni hopes to extend the fundamental perturbation theorem of Kolmogorov, Arnold, and Moser for linear flows on a surface of higher genus. Kolmogorov, Arnold, Moser theory and known results about Arnold diffusion both have significant applications to the problem of the stability of the solar system. These studies deal with Isaac Newton's mathematical model of the solar system and attempt to answer the question as to whether the mathematical model is stable for all time, not just the age of the universe. Hamiltonian systems are a mathematical abstraction of Newton's model of the solar system which applies to many other mechanical systems. Both projects deal with questions in this more general situation related to known results on the stability of the solar systems.