Pendulum type systems from computer algebra viewpoint

We consider periodic solutions to pendulum type ODE systems. The discovery of such solutions is one of the classical problems of mechanics. There exist a variety of methods of computation of periodic solutions, and these methods exist since the formulation of the problems themselves. But these methods were intended for computations by hand, and the attempts to program them for computer algebra systems (CAS) are not always effective. We suggest a means to compute these solutions that is intended for CAS from the beginning. The method is based on application of variational equations of high order and on symbolic differentiation. We demonstrate on a number of examples that all computations are reduced to manipulations with polynomials.