On Periodic Solutions of Autonomous Hamiltonian Systems of Ordinary Differential Equations

For the system <!-- MATH $x''(t) + \operatorname{grad} U(x(t)) = 0$ --> lower bounds are obtained for the number of pairs of odd, periodic solutions, with the period prescribed. These bounds are in terms of the behavior of near the origin and far away from the origin. It is assumed that is even, and two different types of behavior of far away from the origin are considered.