Asymptotic study of the Levitron dynamics

Abstract

In the present effort, we revist the Levitron’s dynamics in the line of previous works due to Berry, Dullin and Easton, and Gans. An invariant set in the Eulerian formulation is delivered and a local study is performed which disclose the dynamics on the invariant manifold which coincides to that obtained by Dullin and Easton using the yaw–pitch–roll angles. Moreover, we extend the results of Gans, being able to determine further stable regions for the magnetic levitation of the Levitron. Symmetric and asymmetric trajectories close to an analytical solution are numerically explored. An asymptotic multiscale analysis is also carried out with the aim of studying the nonlinear interaction between the traslational and rotational modes. By recourse to a Hamiltonian approach, we provide the study of the local behavior of the Levitron near an equilibrium point of the system. Also, the existence of invariant regions in phase space corresponding to persistent levitation of the top are detected. The performed numerical studies serve to elucidate the Levitron’s behavior.