Periodic orbits of conservative systems below the Mañé critical energy value

• Applicant: Professor Dr. Alberto Abbondandolo
• Subject Area: Mathematics
• Term: from 2015 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 273417880

The dynamics of a conservative system is determined by an autonomous Lagrangian on the tangent bundle of the manifold of all configurations. In this proposal, we deal with compact configuration manifolds and Tonelli-type Lagrangians, which define a rather general class of Hamiltonian systems. A fundamental question in conservative dynamics is to understand how the behaviour of the system on an energy level k changes when we let the value of k vary. It is well known that significant changes in this behaviour tend to occur at particular values of the energy, which are known as Mañé critical values. The dynamics above the lower of these values - the Mañé critical value of the universal cover - is reasonably well understood. Much less is known for lower energy levels, in particular when the Lagrangian contains magnetic terms: these terms become dominant for low energies and influence the dynamics in a way which is far from being understood. Here we propose an intensive investigation of these subcritical energy levels.We focus in particular on the questions of existence, multiplicity and linear stability of periodic orbits, as well as orbits satisfying suitable conormal boundary conditions. The methods which we intend to use include variational principles, geometric two-dimensional methods, singular perturbation arguments and symplectic techniques.

• DFG Programme: Research Grants
• International Connection: Brazil, France, United Kingdom
• Cooperation Partners: Professor Dr. Leonardo Macarini; Dr. Marco Mazzucchelli; Professor Dr. Gabriel P. Paternain