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

TL;DR

This paper extends the Poincaré-Cartan integral invariant to nonlinear nonholonomic systems, deriving generalized equations of motion and demonstrating invariance properties that characterize the system's dynamics.

Contribution

It introduces a generalized form of the Poincaré-Cartan integral invariant for nonlinear nonholonomic systems without using Lagrangian multipliers.

Findings

Derived canonical Poincaré-Hamilton equations for nonlinear nonholonomic systems.

Established a version of the Poincaré-Cartan integral invariant involving asynchronous variations.

Showed that the invariance of a line integral characterizes the equations of motion.

Abstract

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.