On an Algorithm for Normalizing an Autonomous Hamiltonian System in the Vicinity of Its Periodic Motion

This paper describes a numerical-analytical algorithm for obtaining the normal form of linear equations of perturbed motion near the time-periodic motion of an autonomous Hamiltonian system with two degrees of freedom. It is assumed that the physical parameters of the material system under study are constant and lie within the region of orbital stability in a first approximation. The algorithm does not use known classical normalization procedures for linear time-periodic Hamiltonian systems. As examples of the calculations required to implement the algorithm, the problem of the motion of a particle on the absolutely smooth surface of a stationary cone in a uniform gravitational field and the problem of the motion of a satellite relative to its center of mass in a circular orbit are considered.
The presented results can serve as the basis for developing a constructive algorithm for obtaining the normal form of nonlinear equations of perturbed motion without resorting to the time-periodic nonlinear canonical transformation that normalizes these equations.