Periodic orbits of conservative systems below the Mañé critical energy value

保守系统的动力学是由所有构型流形的切丛上的自治拉格朗日决定的。在这个方案中，我们处理紧致配置流形和Tonelli型拉格朗日，它们定义了一类相当一般的哈密顿系统。保守动力学中的一个基本问题是理解当我们让k的值变化时，系统在能级k上的行为是如何变化的。众所周知，这种行为的显著变化往往发生在能量的特定值，即所谓的马内临界值。其中较低的值--普遍覆盖的马内临界值--之上的动态是相当好地理解的。对于较低的能级，人们所知的要少得多，特别是当拉格朗日包含磁项时：这些项在低能时占主导地位，并以一种远未被理解的方式影响动力学。在这里，我们建议深入研究这些亚临界能级，特别是周期轨道的存在性、多重性和线性稳定性问题，以及满足适当的余法边界条件的轨道。我们打算使用的方法包括变分原理、几何二维方法、奇异摄动变元和辛技巧。

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.