Construction of analytic solutions of the Cauchy problem for a two-dimensional Hamiltonian system

We consider a two-dimensional Hamiltonian system whose Hamiltonian is the support function of a flat smooth convex compact set that contains the origin in its interior. Closed trajectories of this system are similar to polar curves of the original convex compact set (level lines of the support function). The introduction of generalized polar coordinates reduces the two-dimensional Cauchy problem to a one-dimensional problem, and its solution in some cases can be presented in analytic form. The interest in this topic is connected with the analysis of Chaplygin's generalized problem. We use the technique of support functions; its efficiency in optimal control problems was noted in a number of the authors' papers. Examples are illustrated by graphs.