A class of periodic orbits in classical mechanics (Q1821247)

The existence of periodic solutions to the second-order Hamiltonian system of equations with m degrees of freedom, (1) ẍ\(=-DW(x)\), \(x\in {\mathbb{R}}^ m\), is proved, where DW(x) is the gradient at the point x of the potential function W(x). Also the bifurcation and persistence of unstable periodic orbits of (1) from unstable critical points of W is proved and the results of \textit{R. Churchill, G. Pecelli} and \textit{D. Rod} [ibid. 17, 329-348 (1975; Zbl 0303.34023)] generalized. The applications of the main results in the context of systems with two degrees of freedom conclude the paper.