Poincaré-cartan integral variants and invariants of nonholonomic constrained systems

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