Levitron: multi-scale analysis of stability

Abstract

The Levitron is a typical system for complex Hamiltonian dynamics. Using accurate integrators numerical stability studies were done, especially with respect to variations of the magnetic moment μ for different initial positions x and z. For large μ, equivalent to stronger magnetic fields, the region of stable trajectories is splitted into two parts, whereas for small μ, only one stability region is observed. A linear ansatz is not sufficient to explain this splitting of regions. Vertical and transverse stability conditions have to be combined to understand this behaviour using a multi-scale ansatz. For different hole sizes in the magnetic base plate, the same behaviour appears. For larger holes one has to use stronger magnetic fields. Physically, the stability limits can be identified as critical gradients (forces) of the underlying potential. In x-direction, the stability boundaries are determined by a maximum x-gradient of the potential, which is allowed to act on the top. This derivative of the potential determines a force acting on the top. If this force in x-direction gets too strong, the top deviates from a stable trajectory and gets unstable.