Periodic orbits from generating functions

Periodic orbits are studied using generating functions that yield canonical transformations induced by the phase flow as defined by the Hamilton-Jacobi theory. By posing the problem as a two-point boundary value problem, we are able to develop necessary and sufficient conditions for the existence of periodic orbits of a given period, or going through a given point in space. These conditions reduce the search for periodic orbits to either solving a set of implicit equations, which can often be handled graphically, or to finding the roots of an equation of one variable only. We present an algorithm that solves this problem locally in space for any Hamiltonian dynamical environment. Specific examples of finding periodic orbits in the vicinity of other periodic orbits and around the Libration points in the three-body problem are studied. Introduction Periodic orbits have been widely studied over the last century and are still a topic of great interest. Poincaré already realized their importance for understanding the dynamics of non-integrable Hamiltonian systems when he claimed that they are “the only opening through which we can try to penetrate the stronghold”. Indeed, he conjectured that periodic orbits are dense on typical energy surfaces. Though the Poincaré conjecture is not true for every system (e.g., for a product of harmonic oscillator with incommensurate frequencies), many systems have the property predicted by Poincaré. MacKay provides conditions under which the Poincaré conjecture holds. Many techniques have been developed to find periodic orbits. For instance, in the restricted three-body problem one may use perturbation methods. Such a method allows one to find families of periodic orbits very efficiently, but does not provide a systematic procedure to find a periodic orbit of either a given period or going through a given point. By using the generating functions from the Hamilton-Jacobi theory, we can solve such a problem. Indeed, we can reduce the search for periodic orbits to either solving a set of implicit equations, which can often be done graphically, or to finding the roots of an equation of one variable only. The method we propose is independent of the Hamiltonian system considered. Nevertheless, this method is only suitable for systems for which the generating functions are known, that is, for which we can solve the Hamilton-Jacobi equation. This equation is usually very hard to solve. Some algorithms have been developed, but each of them has its own peculiarities. In this paper, we will be using the algorithm developed by Guibout and Scheeres. It allows us to compute the generating functions for any Hamiltonian system describing nonlinear relative motion of particles moving in a Hamiltonian field. Hence, we will be able to study periodic orbits in the vicinity of any trajectory, and will be focusing on periodic orbits in the vicinity of other periodic orbits and around the Libration points in the three-body problem. ∗Copyright c ©2003 The American Astronautical Society. †Graduate Research Assistant, PhD Candidate, Aerospace Engineering Department, FXB Building, 1320 Beal Avenue, Ann Arbor, MI 48109-2140, [email protected] ‡Associate Professor, Senior member AIAA, Aerospace Engineering Department, FXB Building, 1320 Beal Avenue, Ann Arbor, MI 48109-2140, [email protected]