Stability from index estimates for periodic solutions of Lagrangian systems (Q1320283)

The Lagrangian system of the form (1) \(\ddot x+ \nabla_ x U(x,t)= 0\), \(x\in \mathbb{R}^ n\), \(t\in \mathbb{R}\), where the potential energy function \(U\) is periodic in \(t\) of period \(T\), is considered. It is assumed that \(U\) is the sum of a positive quadratic term and of a term \(V\) (super- quadratic or subquadratic, convex or concave, uniformly in \(t\)). Several theorems on multiplicity, minimality of period and stability for periodic solutions of the Lagrangian system (1) are proved.