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

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