Prescribed minimal period problems for convex Hamiltonian systems via Hofer-Zehnder symplectic capacity (Q5928274)

Let \(H\) be an autonomous Hamiltonian convex in both space and momenta coordinates. The energy surfaces \(H=e\) bound convex domains in \(\mathbb{R}^n\). The Hofer-Zehnder capacity \(c(e)\) of these domains is a symplectic invariant introduced in [Analysis et cetera, Res. Pap. in Honor of J. Moser's 60th Birthday, 405-427 (1990; Zbl 0702.58021)], which provides a more direct way to obtain important results in symplectic topology [\textit{H. Hofer} and \textit{E. Zehnder}, Symplectic invariants and Hamiltonian dynamics, Birkhäuser, Basel (1994; Zbl 0805.58003)]. The subject of this paper is the behaviour of \(c(e)\) as a function of \(e\), more precisely the differentiability of \(c(e)\). The main consequence is that a convex autonomous Hamiltonian system will have periodic solutions with any prescribed minimal period in the interval \([c'(a),c'(b)]\) as long as \(c'(a)\leq c'(b)\). Applications are given for: asymptotically quadratic Hamiltonian systems, subquadratic Hamiltonian systems and classical systems with subquadratic potentials.