Poincaré recurrence and number theory

Abstract

Poincar is largely responsible for the transformation of celestial mechanics from the study of individual solutions of differential equations to the global analysis of phase space. A system of differential equations such as those which embody the laws of Newtonian mechanics generates a one-parameter group of transformation of the manifold that represents the set of states of a dynamical system. The evolution of the dynamical system in time corresponds to a particular solution of the system of differential equations; it also corresponds to an orbit of the group of transformations acting on a single state. The efforts of the classical analysts in celestial mechanics had been directed to extracting by analytical means as much information as possible about the individual solutions to the system of differential equations. Poincar's work gave impetus to a global approach which studies the totality of solutions and shifts attention to the transformation group of phase space.