Geodesic motion on the symplectic leaf of $SO(3)$ with distorted $e(3)$ algebra and Liouville integrability of a free rigid body

Abstract

The solutions to the Euler-Poisson equations are geodesic lines of $SO(3)$ manifold with the metric determined by the inertia tensor. However, the Poisson structure on the corresponding symplectic leaf does not depend on the inertia tensor. We calculate its explicit form and confirm that it differs from the algebra $e(3)$. The obtained Poisson brackets are used to demonstrate the Liouville integrability of a free rigid body. The general solution to the Euler-Poisson equations is written in terms of exponential of the Hamiltonian vector field.

6 pages, the title slightly changed, discussions added. Matches with published version. A typo in Eq. (32) has been corrected

References in corpus (2)

• Massless polarized particle and Faraday rotation of light in the Schwarzschild spacetime
• Basic notions of Poisson and symplectic geometry in local coordinates, with applications to Hamiltonian systems

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