Euler-Poisson equations of a dancing spinning top, integrability and examples of analytical solutions

TL;DR

This paper derives equations for a dancing top, explores their integrability, and provides analytical solutions for specific cases, highlighting the effects of constraints on the system's mathematical structure.

Contribution

It introduces a novel formulation of dancing top equations from a variational principle and presents explicit analytical solutions for symmetrical cases.

Findings

Analytical solutions for symmetrical dancing top with fixed center of mass height

Modified Poisson structure due to support reaction constraints

Questioning of Liouville integrability under additional constraints

Abstract

Equations of a rotating body with one point constrained to move freely on a plane (dancing top) are deduced from the Lagrangian variational problem. They formally look like the Euler-Poisson equations of a heavy body with fixed point, immersed in a fictitious gravity field. Using this analogy, we have found examples of analytical solutions for the case of a heavy symmetrical dancing top. They describe the motions with center of mass keeping its height fixed above the supporting plane. General solution to equations of a dancing top in terms of exponential of Hamiltonian field is given. An extra constraint, that take into account the reaction of supporting plane, leads to modification of the canonical Poisson structure and therefore the integrability according to Liouville is under the question.