We study the periodic orbits problem on energy levels of Tonelli Lagrangian systems over configuration spaces of arbitrary dimension. We show that, when the fundamental group is finite and the Lagrangian has no stationary orbit at the Mane critical energy level, there is a waist on every energy level just above the Mane critical value. With a suitable perturbation with a potential, we show that there are infinitely many periodic orbits on every energy level just above the Mane critical value, and on almost every energy level just below. Finally, we prove the Tonelli analogue of a closed geodesics result due to Ballmann-Thorbergsson-Ziller.

中文翻译：

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高维 Tonelli 系统的腰部定理

我们研究了任意维度配置空间上托内利拉格朗日系统能级的周期轨道问题。我们表明，当基本群有限且拉格朗日在 Mane 临界能级没有静止轨道时，在 Mane 临界值上方的每个能级上都有一个腰部。通过具有合适的势能扰动，我们表明在刚好高于 Mane 临界值的每个能级上以及几乎在刚好低于每个能级上的每个能级上都有无限多的周期轨道。最后，我们证明了由于 Ballmann-Thorbergsson-Ziller 的封闭测地线结果的 Tonelli 模拟。