Classification of the classical $SL(2,\mathbb{R})$ gauge transformations in the rigid body

TL;DR

This paper revisits the classification of gauge transformations in the Euler top system using Nambu's bi-Hamiltonian framework, explicitly describing Hamiltonian vector fields and classifying Lie-Poisson structures, including symmetric and asymmetric cases.

Contribution

It provides an explicit form of Hamiltonian vector fields for all Lie-Poisson structures and offers a comprehensive classification of these structures in the Euler top system.

Findings

Explicit Hamiltonian vector fields for all Lie-Poisson structures

Complete classification of Lie-Poisson structures in the Euler top

Recovery of previously reported structures in the literature

Abstract

In this paper we revisit the classification of the gauge transformations in the Euler top system using the generalized classical Hamiltonian dynamics of Nambu. In this framework the Euler equations of motion are bi-Hamiltonian and $SL(2, \mathbb{R})$ linear combinations of the two Hamiltonians leave the equations of motion invariant, although belonging to inequivalent Lie-Poisson structures. Here we give the explicit form of the Hamiltonian vector fields associated to the components of the angular momentum for every single Lie-Poisson structure including both the asymmetric rigid bodies and its symmetric limits. We also give a detailed classification of the different Lie-Poisson structures recovering all the ones reported previously in the literature.