$S^1$-索引理论用于洛伦兹力方程
$\bf{S^1}$-index theory for the Lorentz force equation
AI总结：
本文通过$S^1$-索引理论证明了洛伦兹力方程存在多个周期解，方法基于Lusternik-Schnirelman方法与弱紧致条件。
摘要：
在本文中，我们证明了与洛伦兹力方程相关的庞加莱作用泛函的$S^1$-不变性导致存在多个临界点，这些临界点是具有固定周期的周期解。为此，我们证明了一个抽象的多重性结果，其基于Lusternik-Schnirelman方法与$S^1$-索引。在Fadell-Rabinowitz索引背景下相应结果已在Ekeland和Lasry（Ann. Math., 112 (1980)）中证明。我们抽象结果的主要特点是它允许考虑满足仅弱紧致条件的非光滑泛函，该条件适合于庞加莱泛函。
英文摘要：
In this paper we prove that the $S^1$-invariance of the Poincaré action functional associated to the Lorentz force equation gives the existence of multiple critical points which are periodic solutions with a fixed period. To do this, we prove an abstract multiplicity result which is based upon the Lusternik-Schnirelman method with the $S^1$-index. The corresponding result in the context of the Fadell-Rabinowitz index is proved in Ekeland and Lasry (Ann. Math., 112 (1980)). The main feature of our abstract result is that it allows us to consider nonsmooth functionals satisfying only a weak compactness condition well adapted to the Poincaré functional.
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