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

Abstract

AbstractIn 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 $${\mathbb {S}}^2$$ S 2 , where $$K: {\mathbb {S}}^2 \rightarrow {\mathbb {R}}$$ K : S 2 → 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].