A Generalization of the Poincar\'e-Cartan Integral Invariant for a Nonlinear Nonholonomic Dynamical System

Based on the d’Alembert-Lagrange-Poincar'{e} variational principle, we formulate general equations of motion for mechanical systems subject to nonlinear nonholonomic constraints, that do not involve Lagrangian undetermined multipliers. We write these equations in a canonical form called the Poincar'{e}-Hamilton equations, and study a version of corresponding Poincar'{e}-Cartan integral invariant which are derived by means of a type of asynchronous variation of the Poincar'{e} variables of the problem that involve the variation of the time. As a consequence, it is shown that the invariance of a certain line integral under the motion of a mechanical system of the type considered characterizes the Poincar'{e}-Hamilton equations as underlying equations of the motion. As a special case, an invariant analogous to Poincar'{e} linear integral invariant is obtained.