Poincar\'e-Chetaev equations in the Dirac's formalism of constrained systems

TL;DR

This paper introduces a formalism for constrained systems on group manifolds that simplifies the Poisson structure construction, exemplified by deriving Poincaré-Chetaev equations on the SO(3) manifold.

Contribution

It presents a novel approach to handle non-canonical coordinates on group manifolds, avoiding explicit Dirac bracket calculations, and applies it to derive Poincaré-Chetaev equations.

Findings

Simplified construction of Poisson structures on group manifolds.

Derivation of Poincaré-Chetaev equations for SO(3).

Explicit solution expressed via exponential of Hamiltonian vector field.

Abstract

We single out a class of Lagrangians on a group manifold, for which one can introduce non-canonical coordinates in the phase space, which simplify the construction of the Poisson structure without explicitly calculating the Dirac bracket. In the case of $SO(3)$\,- manifold, the application of this formalism leads to the Poincar\'e-Chetaev equations. The general solution to these equations is written in terms of exponential of the Hamiltonian vector field.