Homoclinics and heteroclinics for a Lagrangian system
The paper considers a family of Lagrangian systems on \(\mathbb{T}^{2}\), described by the Lagrangian \[ L(q)=\sum_{i,j=1}^{2}a_{ij}(q)\dot q_{i}\dot q_{j}-V(q), \] where \(V\) and \(a_{i,j}\) are \(C^{2}\)-differentiable and \(V(0)=0>V(x)\), \(x\in \mathbb{T}^{2}\). In recent years some papers [see e.g. \textit{S. Bolotin} and \textit{P. Negrini}, A variational criterion for nonintegrability, Russ. J. Math. Phys. 5, 415-436 (1998)] have established the existence of heteroclinic orbits of prescribed homotopy type for such families. The results required a geometrical assumption that implies the minimum of an associated variational problem is a periodic orbit of the Lagrangian system rather than a homoclinic on \(\mathbb{T}^{2}\). In this paper the author shows that some stronger geometrical condition together with the variational framework developed by the author leads to much richer families of homoclinic and heteroclinic solutions of the system.
scientific article; zbMATH DE number 1370122