The principle of least action and periodic solutions in problems of classical mechanics: PMM vol. 40, n≗ 3, 1976, pp. 399–407

Abstract

The problem of existence of periodic solutions of equations of natural mechanical system motions is considered in the case when region D of all possible motions is bounded. Periodic libration solutions are derived for systems with many degrees of freedom. The trajectory of such solution is diffeomorphic to segment [0, 1], its ends lie at the boundary of D, and the representative point oscillates along that curve. Existence of libration solutions is proved in the case when the region of possible motions is diffeomorphic to the direct product N × [0, 1], where N is a smooth compact manifold. Obtained results are applied in the problem of motion of a solid body with a fixed point in a Newtonian force field.