Poincaré–Chetaev Equations in 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 the SO(3) manifold, the application of this formalism leads to the Poincaré–Chetaev equations. The general solution to these equations is written in terms of an exponential of the Hamiltonian vector field.

Improved Equations of the Lagrange Top and Examples of Analytical Solutions

“…The last equation from (38) implies that besides the integrals of motion (18) and ( 19), there is one more: Ω 3 = const. This can also be seen from Equation (26). Indeed, the third component of this equation reads as follows:…”

“…Equations ( 23) and ( 24) represent a Hamiltonian system [23][24][25]. This can be confirmed by constructing the Hamiltonian formulation of the Lagrangian theory (20) with the help of intermediate formalism developed in [26]. This gives the Hamiltonian…”

“…Their formal solution can be written in terms of the exponential of the Hamiltonian vector field; see [26]. For the centre of mass vector z(0) in a general position, the solution to these equations in quadratures is not known.…”