On periodic solutions of autonomous Hamiltonian systems of ordinary differential equations

Abstract

For the system x ( t ) + grad ⁡ U ( x ( t ) ) = 0 x(t) + \operatorname {grad} U(x(t)) = 0 lower bounds are obtained for the number of pairs ± x ( t ) \pm x(t) of odd, periodic solutions, with the period prescribed. These bounds are in terms of the behavior of U ( x ) U(x) near the origin and far away from the origin. It is assumed that U ( x ) U(x) is even, and two different types of behavior of U ( x ) U(x) far away from the origin are considered.