Quasiperiodic solutions of variational problems of motion in a central force field

Abstract

A method is proposed for computing nearly optimal trajectories of dynamic systems with a small parameter by splitting the original variational problem into two separate problems for "fast" and "slow" variables. The problem for "fast" variables is solved by improving the zeroth approximation — the extremals of the linearized problem — by the Ritz method. The solution of the problem for "slow" variables is constructed by passing from a discrete argument — the number of revolutions around the attracting center— to a continuous argument. The proposed method does not require numerical integration of systems of differential equations and produces a highly accurate approximate solution of the problem.