  We consider the compatibility of the equations of motion which follow from
d'Alembert's principle in the case of a general autonomous non-holonomic
mechanical system in N dimensions, with those equations which follow for the
same system by assuming the validity of a specific variational action
principle, in which the non-holonomic conditions are implemented by means of
the multiplication rule in the calculus of variations. The equations of motion
which follow from the principle of d'Alembert are not identical to the
equations which follow from the variational action principle. We give a proof
that the solutions to the equations of motion which follow from d'Alembert's
principle do not in general satisfy the equations which follow from the action
principle with non-holonomic constraints. Thus the principle of d'Alembert and
the minimal action principle involving the multiplication rule are not
compatible in the case of systems with non-holonomic constraints. For
simplicity the proof is given for autonomous systems only, with one general
non-holonomic constraint, which is linear in the generalized velocities of the
system.
