Usually there does not exist an integral invariant of Poincare-Cartan's typ
e for a nonholonomic system because a constraint submanifold does not admit
symplectic structure in general. An integral variant of Poincare-Cartan's
type, depending on the nonholonomy of the constraints and nonconservative f
orces acting on the system, is derived from D'Alembert-Lagrange principle.
For some nonholonomic constrained mechanical systems, there exists an alter
native Lagrangian which determines the symplectic structure of a constraint
submanifold. The integral invariants can then be constructed for such syst
ems.