The minimal period problem of classical Hamiltonian systems with even potentials
The author investigates the existence of non-constant periodic solutions with prescribed minimal period for a conservative (the author says ``second order Hamiltonian'') system of the form \(\ddot x + \text{grad }V = 0\) in \(\mathbb{R}^ n\), where the potential \(V: \mathbb{R}^ n \to \mathbb{R}\) is an even superquadratic autonomous function, with no convexity assumptions. He observes that the usual direct variational formulation of the system possesses a natural \(V_ 4\)-symmetry \((V_ 4 = Z_ 2 \oplus Z_ 2\) is the Kleinian Fourgroup). Using this as well as a new iteration inequality on the Morse index and Mountain-pass theorem, the author shows that for every \(T > 0\) the above system possesses a \(T\)-periodic solution with minimal period \(T\) or \(T/3\) and this solution is even about \(t = 0\), \(t = T/2\) and odd about \(t = T/4\), \(t = 3T/4\).
scientific article; zbMATH DE number 554341
scientific article