Many closed $K$-magnetic geodesics on ${\mathbb{S}}^2$

Abstract

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 $K:{\mathbb{S}}^2\to \mathbb{R}$ 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.

27 pages. To appear in Manuscripta Mathematica