The stability of the periodic solutions of second order Hamiltonian systems (Q1567596)

This paper is devoted to the stability of the periodic solutions of the following second order Hamiltonian systems \[ \ddot x+ V_x'(t,x)= 0, \quad x\in \mathbb{R}^n, \tag{1} \] where \(n\) is a positive integer, \(V: \mathbb{R}\times \mathbb{R}^n\to \mathbb{R}\) is a function which is \(\tau\)-periodic in the variable \(t\). \(V_x'\) denotes its gradient with respect to \(x\). Under some natural conditions on \(H\) the author analyzes two different cases: superquadratic and subquadratic cases. He proves that in the subquadratic case, there exist infinitely many geometrically distinct elliptic periodic solutions, and in the superquadratic case, there exist infinitely many geometrically distinct periodic solutions with at most one instability direction if they are half period non-degenerated, otherwise they are elliptic.