Existence of T-periodic solutions for a class of Lagrangian systems (Q5966358)

We study the following Lagrangian system of differential equations: \[ (*)\quad \frac{d}{dt}\frac{\partial {\mathcal L}}{\partial \xi}(q,\dot q,t)- \frac{\partial {\mathcal L}}{\partial q}(q,\dot q,t)=0,\quad q\in C^ 2({\mathbb{R}},{\mathbb{R}}^ N) \] where \({\mathcal L}\) denotes the Lagrangian function \[ {\mathcal L}(q,q,t)=\frac{1}{2}\sum^{N}_{i,j=1}a_{ij}(q)\xi_ i\xi_ j-V(q- t),\quad q,\xi \in {\mathbb{R}}^ N,\quad t\in {\mathbb{R}}. \] First we state the existence of multiple periodic solutions of prescribed period when the potential V is unbounded and subquadratic at infinity. Then we prove the existence of at least one nontrivial solution of problem (*) when the potential \(V=V(q,t)\) is bounded and T-periodic. Finally we consider the following forced Lagrangian system: \[ (**)\quad \frac{d}{dt}\frac{\partial {\mathcal L}}{\partial \xi}(q,\dot q)- \frac{\partial {\mathcal L}}{\partial q}(q,\dot q)=g(t) \] where g is a T- periodic function; we prove the existence of at least one T-periodic solution of problem (**) when the potential is unbounded and subquadratic at infinity. The results contained in this paper are obtained using variational methods and generalize some well known results referred to the case of the \(a_{ij}'s\) constant.