Top on a smooth plane

Abstract

We investigate the dynamics of a sliding top that is a rigid body with an ideal sharp tip moving in a perfectly smooth horizontal plane, so no friction forces act on the body. We prove that this system is integrable only in two cases analogous to the Euler and Lagrange cases of the classical top problem. The cases with the constant gravity field with acceleration $g\neq0$ and without external field $g=0$ are considered. The non-integrability proof for $g\neq0$ based on the fact that the equations of motion for the sliding top are a perturbation of the classical top equations of motion. We show that the integrability of the classical top is a necessary condition for the integrability of the sliding top. Among four integrable classical top cases the corresponding two cases for the sliding top are also integrable, and for the two remaining cases, we prove their non-integrability by analyzing the differential Galois group of variational equations along a certain particular solution. In the absence of constant gravitational field $g=0$ the integrability is much more difficult. At first, we proved that if the sliding top problem is integrable, then the body is symmetric. In the proof, we applied one of the Ziglin theorem concerning the splitting of separatrices phenomenon. Then we prove the non-integrability of the symmetric sliding top using differential Galois group of variational equations except two the same as for $g\neq0$ cases. The integrability of these cases is also preserved when we add to equations of motion a gyrostatic term.