A Generalization of the Poincaré-Cartan Integral Invariant for a Nonlinear Nonholonomic Dynamical System

Details A Generalization of the Poincaré-Cartan Integral Invariant for a Nonlinear Nonholonomic Dynamical System A Generalization of the Poincaré-Cartan Integral Invariant for a Nonlinear Nonholonomic Dynamical System

2007-09-17

arxiv.org/abs/0709.2523v2

Physics

Mathematical Physics

Scientific paper

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.

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