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

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.