Geometry and physics of Arnold diffusion

Friday, January 27, 2012 - 16:00 to 16:50
Speaker Information
Mark Levi
Penn State University

Abstract or Additional Information

Many physical systems, such as the Solar system, for instance, can be considered as small perturbations of completely integrable Hamiltonian systems. In the case of the Solar system the interaction between planets can be viewed as a small perturbation of the otherwise decoupled system of Kepler problems. Kolmogorov-Arnold-Moser theory, arguably the main development in Hamiltonian dynamics of the past century and a half, shows that small perturbations do not affect qualitative behavior of the majority of solutions of a completely integrable Hamiltonian system (under some general assumptions). This majority forms a Cantor set in the phase space. Arnold diffusion is the phenomenon that is believed to happen in the complement to this "surviving" set of solutions, for a generic Hamiltonian system.

In this talk I will give an intuitive geometrical illustration of this phenomenon for geodesics on the torus, or, equivalently, for the motion of a particle in a periodic potential. With this interpretation, the concepts such as resonances, the "whiskered tori" and the heteroclinic orbits acquire a simple geometrical meaning. At the end, I will show how Arnold diffusion can manifest itself in a chain of coupled pendula as the slow "leakage" of energy from one pendulum to its neighbor. This leakage can happen in a random-looking fashion, i.e. along an arbitrarily prescribed itinerary.

Research Area