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Based on the d'Alembert-Lagrange-Poincar\'{e} variational principle, weformulate general equations of motion for mechanical systems subject tononlinear nonholonomic constraints, that do not involve Lagrangian undeterminedmultipliers. We write these equations in a canonical form called thePoincar\'{e}-Hamilton equations, and study a version of correspondingPoincar\'{e}-Cartan integral invariant which are derived by means of a type ofasynchronous variation of the Poincar\'{e} variables of the problem thatinvolve the variation of the time. As a consequence, it is shown that theinvariance of a certain line integral under the motion of a mechanical systemof the type considered characterizes the Poincar\'{e}-Hamilton equations asunderlying equations of the motion. As a special case, an invariant analogousto Poincar\'{e} linear integral invariant is obtained.

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