In this paper we adopt an alternative, analytical approach to Arnol’d problem [4] about the existence of closed and embedded K-magnetic geodesics in the round 2-sphere \({\mathbb {S}}^2\), where \(K: {\mathbb {S}}^2 \rightarrow {\mathbb {R}}\) is a smooth scalar function. In particular, we use Lyapunov-Schmidt finite-dimensional reduction coupled with a local variational formulation in order to get some existence and multiplicity results bypassing the use of symplectic geometric tools such as the celebrated Viterbo’s theorem [21] and Bottkoll results [7].

中文翻译：

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$$ {\ mathbb {S}} ^ 2 $$ S 2上的许多闭合K磁测地线

在本文中，我们对圆形2球\（{\ mathbb {S}} ^ 2 \）中存在封闭和嵌入的K磁大地测量学的Arnol问题[4]采用了一种替代的分析方法，其中\（K：{\ mathbb {S}} ^ 2 \ rightarrow {\ mathbb {R}} \）是一个光滑的标量函数。特别是，我们使用Lyapunov-Schmidt有限维归约法和局部变分公式，以绕过使用诸如著名的维特博定理[21]和Bottkoll结果[7]之类的几何工具来获得一些存在性和多重性结果。