  The aim of this paper is to show that the Lagrange-d'Alembert and its
equivalent the Gauss and Appel principle are not the only way to deduce the
equations of motion of the nonholonomic systems. Instead of them, here we
consider the generalization of the Hamiltonian principle for nonholonomic
systems with nonzero transpositional relations.
  By applying this variational principle which takes into the account
transpositional relations different from the classical ones we deduce the
equations of motion for the nonholonomic systems with constraints that in
general are nonlinear in the velocity. These equations of motion coincide,
except perhaps in a zero Lebesgue measure set, with the classical differential
equations deduced with d'Alembert-Lagrange principle.
  We provide a new point of view on the transpositional relations for the
constrained mechanical systems: the virtual variations can produce zero or
non-zero transpositional relations. In particular the independent virtual
variations can produce non-zero transpositional relations. For the
unconstrained mechanical systems the virtual variations always produce zero
transpositional relations.
  We conjecture that the existence of the nonlinear constraints in the velocity
must be sought outside of the Newtonian model.
  All our results are illustrated with precise examples.
