This article investigates the dynamics of a sliding top, a rigid body with a sharp tip moving on a frictionless horizontal plane. It proves that the system is integrable only in two cases analogous to the Euler and Lagrange cases of the classical top problem, considering scenarios with and without a constant gravity field. The non-integrability proof for a non-zero gravity field is based on the fact that the equations of motion for the sliding top are a perturbation of the classical top equations of motion. It is shown that the integrability of the classical top is a necessary condition for the integrability of the sliding top. In the absence of a constant gravitational field, the integrability is much more difficult. First, we proved that if the sliding top problem is integrable, then the body is symmetric. In the proof, we applied the Ziglin theorem concerning the splitting of separatrices phenomenon. Then, we prove the non-integrability of the symmetric sliding top using the differential Galois group of variational equations except two the same as for g≠0 cases. The integrability of these cases is also preserved when we add to equations of motion a gyrostatic term.
As a theoretical analysis of a dynamic system, this article is aligned with the Chaos: An Interdisciplinary Journal of Nonlinear Science's scope on nonlinear dynamics and complex systems. The study contributes to the theoretical understanding of rigid body motion and integrability, fitting within the journal’s focus on mathematical and physical aspects of chaotic systems.