Hierarchy of Periodic Solutions for Hamiltonian Systems and their Applications to Celestial Mechanics

Abstract

AbstractA systematic investigation has been carried out for periodic solutions for standard-form Hamiltonian systems containing a small parameter/the principal problem of dynamics/. An efficient method of investigation of conditions for periodicity of solutions has been developed. Besides fitting the initial conditions of the action-angle variables, the idea of fitting the values of the parameters of the problem is used. Constructive conditions are obtained for the existence of periodic solutions in both principal and degenerate cases, as well as necessary conditions for their stability; algorithms have been developed for constructing these solutions as series in integer powers of the small parameter. To study particular periodic solutions /by high order resonances/, canonical transformations of the initial equations to a special form are used.