Periodic solutions with prescribed minimal period for convex autonomous Hamiltonian systems (Q1077035)

In 1977, Rabinowitz proved the existence of a nonconstant periodic solution of any prescribed period \(T>0\) for the Hamiltonian systems under some reasonable conditions. However, he does not claim that the T- periodic solution he finds actually has minimal period T. Therefore, it challenges one's attention to prove minimality of the period for that T- periodic solution. The first researchers to make progress on this question were A. Ambrosetti and G. Mancini, and then M. Girardi and M. Matzeu. This paper is a similar one devoted to the same question for convex autonomous Hamiltonian systems with some (more or less) weaker conditions. The authors combine two new tools: An index theory for periodic solutions of Hamiltonian systems, and a complete description of \(C^ 2\) functionals near critical points of mountain pass type. The main result is that the Ambrosetti-Rabinowitz theorem, when applied to the dual action functional, will actually yield solutions with minimal period T.