The nonholonomic bracket on contact mechanical systems

Abstract

In this paper, we study contact nonholonomic mechanical systems. The contribution of the paper could be divided into two blocks. Firstly, we present a general framework for contact constrained dynamics in such a way that the constraints are geometrically described by a submanifold of the contact manifold and the reaction forces are given by a distribution along the constraint submanifold. In this framework, we describe the constrained differential equations, examine the conditions for the existence and uniqueness of solutions of these equations, and construct two different brackets of functions which describe the evolution of the system. We also prove that nonholonomic contact Lagrangian systems are particular cases of the above general framework. In addition, this general framework permits us to develop the Hamiltonian counterpart and, in this setting, we present the second main contribution of the paper: the construction of another bracket, which is a natural extension of that defined by R.J. Eden, but now it is an almost Jacobi bracket since it does not satisfy the Leibniz rule.