Equations of Motion of Nonholonomic Hamiltonian Systems

We obtain the equations of motion for a Lagrangian dynamical system under nonholonomic constraints making use of the D’Alembert principle. We show that the Lagrange multipliers can be expressed in terms of the Poisson bracket of the Hamiltonian and the constraint. This appealing result greatly simplifies the derivation of the equations of motion. The existence of at least two conserved quantities; the Hamiltonian evaluated on the constrained manifold and the constraint itself, can be readily shown from the expression of the Lagrange multipliers. The method is verified with several examples found in the literature derived with more sophisticated machinery as the nonholonomic brackets.