$\mathbb S^2$上的许多闭合$K$ -磁测地线
Many closed $K$-magnetic geodesics on $\mathbb S^2$
摘要：
本文对Arnol'd问题\cite{A1}采用了另一种分析方法，该问题是关于在圆形$2$球$\mathbb S^2$中存在封闭的嵌入式$K$ -磁性测地线，其中$K: \mathbb S^2 \rightarrow \mathbb R$是光滑的标量函数.特别是，我们使用Lyapunov-Schmidt有限维约化加上局部变分公式，以获得一些存在性和多重性的结果绕过使用辛几何工具，如著名的Viterbo定理和Bottkoll结果。
英文摘要：
In this paper we adopt an alternative, analytical approach to Arnol'd problem \cite{A1} 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 and Bottkoll results.
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