A method for continuing families of periodic solutions of Lagrangian systems

Abstract

An autonomous multidimensional Lagrangian system of differential equations depending on parameters is considered. A predictor-corrector method is proposed for constructing a family of periodic solutions (including retrograde solutions) obtained from a given solution by variation of the parameters. In the context of the wide range of problems in classical and celestial mechanics that are described by Lagrangian systems of differential equations, it is particularly interesting to study non-isolated periodic solutions of such equations, parametrized by both extrinsic and intrinsic parameters (the latter are represented by the initial conditions of the solution, such as the energy constant). A family parametrized by the energy constant is known as a natural family /1, 2/. The classical example of a natural family of periodic orbits is provided by the Lyapunov periodic motions /3/ originating from the equilibrium position of a Hamiltonian system. Existing methods of investigating them /4, 5/ are based on the introduction of local coordinates in the neighbourhood of a periodic solution and subsequent normalization of the equations of the perturbed motion, and the construction of the solution as a series with respect to a small parameter, where the latter characterizes the deviation of the motion from the equilibrium position. In a more complicated situation the generating solution is known only in terms of its initial conditions and period, the solution itself being obtained by numerical integration of the original system. Consequently, any method for continuing a (not necessarily Lyapunov) family by expansion with respect to the parameters is necessarily numerical. On the available variety of such methods we mention a series of papers by Sarychev and Sazonov (for references see /6/), who have worked out a highly efficient method for solving the boundary-value problems that arise in this context. This paper draws on ideas similar to those of /7–9/ ∗∗, (∗∗See also Sokol'skii A.G. and Khovanskii S.A., A computation algorithm for continuing periodic solutions of two-dimensional Hamiltonian systems as functions of the parameters, Moscow, MAI, 1986. Deposited at VINITI, 4.05.86, 4042-86.) where a predictor-corrector method for continuation of periodic solutions is worked out for systems with two degrees of freedom. This method will be generalized here to multidimensional systems, for which Birkhoff 's theorem about the reduction of a two-dimensional system to canonical form is no longer valid; it is nevertheless possible to reduce the boundary-value problem to a Cauchy problem by a special choice of local coordinates.