THE GRADIENTIZATION OF CONSTRAINED HERGLOTZ EQUATIONS AND THE STABILITY OF THEIR SOLUTIONS FOR NONHOLONOMIC SYSTEMS

Herglotz's generalized variational principle is a generalization of Hamilton's principle for nonconservative systems. It establishes variational descriptions of nonconservative processes by defining functional functions of differential equations. Constrained Herglotz equations are a kind of dynamic equations of nonconservative systems with nonholonomic constraints derived from Herglotz principle combined with Appell-Cheatev conditions on virtual displacements, which provides a new perspective for the study of nonconservative problems. Gradient system is a system of differential equations, which has many good properties and is suitable to be studied by Lyapunov function. If a mechanical system can be reduced to a gradient system, the latter can be used to study its dynamic behavior. In this paper, the gradientization of constrained Herglotz equations is proposed, and the stability of their solutions is studied. Firstly, the constrained Herglotz equations for nonholonomic systems are given and expressed in contravariant algebraic form. Secondly, four kinds of basic gradient systems are introduced and their equations are set out. Thirdly, the gradientization of constrained Herglotz equations is studied, and the sufficient conditions to change constrained Herglotz equations into four kinds of basic gradient systems are established. Obviously, if the conditions are not satisfied, it cannot be determined that the equation is not a gradient system. Fourth, the stability of the solutions of constrained Herglotz equation are studied. Finally, four examples of nonconservative nonholonomic systems are given, which are reduced to four types of basic gradient systems, and the stability of their solutions is studied. The examples show the effectiveness of the method and the results. This paper provides a method to determine the stability of a class of nonconservative systems subject to nonholonomic constraints.