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

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.

arXiv comment12 pages, typos fixed

References in corpus (8)

• Motion of charged and spinning particle influenced by dark matter field surrounding a charged dyonic black hole
• Gravitational Faraday rotation of gravitational waves by a Kerr black hole
• Pseudogauge freedom and the SO(3) algebra of spin operators
• Lagrangian and Hamiltonian formulations of asymmetric rigid body, considered as a constrained system
• Geodesic motion on the symplectic leaf of $SO(3)$ with distorted $e(3)$ algebra and Liouville integrability of a free rigid body
• Basic notions of Poisson and symplectic geometry in local coordinates, with applications to Hamiltonian systems
• Collisional Penrose process with spinning particles in braneworld black hole
• Coulomb problem for classical spinning particle

Cited by in corpus (5)

• An asymmetrical body: example of analytical solution for the rotation matrix in elementary functions and Dzhanibekov effect
• General solution to Euler-Poisson equations of a free symmetric body by direct summation of power series
• Lagrange top: integrability according to Liouville and examples of analytic solutions
• Dynamics of a relativistic discrete body: rigidity conditions, and covariant equations of motion
• An alternative interpretation of the Grioli gyroscope suspension points