On periodic solutions of autonomous Hamiltonian systems of ordinary differential equations

Clark, David C.

doi:10.1090/S0002-9939-1973-0315217-8

On periodic solutions of autonomous Hamiltonian systems of ordinary differential equations
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Proc. Amer. Math. Soc. 39 (1973), 579-584 Request permission

Abstract:

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

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