On Poincarje-Chetaev's equations

Valentin V. Rumyantsev

The remarkable idea of II. Poincare to represent equations of motion of a holonomic system by means of some transitive Lie's group of infinitesimal transformations was generalized by N. Chetaev for the case of non-stationary constraints and of dependent variables with an intransitive group of virtual displacements. Chetaev has also proposed the canonical form and the generalized Hamilton-Jacobi equation for Poincare’s equations and proved the generalizations of the Poisson and Jacobi theorems. In the lecture I prove that in the general case the Poincare-Chetaev’s canonical equations are the Hamiltonian equations for the non-canonical formulation. It is also proved that the equations of motion of a system written in superfluous coordinates and that of Euler-Lagrange written in quasi-coordinates are the special forms of the Poincare-Chetaev’s equations. The question of using the Poincare-Chetaev’s equations for non-holonomic systems is also discussed.