ALGEBRAIC STRUCTURE AND POISSON THEORY OF HERGLOTZ EQUATIONS FOR NONCONSERVATIVE SYSTEMS

The equations of holonomic or nonholonomic constrained systems generally do not have a Lie algebraic structure if there are nonconservative forces, so the classical Poisson integral theory can only be partially applied to the integral problems of these systems. A class of dynamic equations, i.e., Herglotz’s equations, for nonconservative systems can be derived by Herglotz principle. Studying the algebraic structure of Herglotz equation and establishing its Poisson integral theory is of great significance for exploring the dynamics of nonconservative systems. In this paper, the algebraic structure of Herglotz equations is studied, and then Poisson theory for nonconservative systems is established, including holonomic and nonholonomic cases. Firstly, the Herglotz equations for holonomic nonconservative systems are established, and the equations are transformed into contravariant algebraic form by introducing integral factor. The equations are proved to have Lie algebraic structure, and the Poisson theory can be applied to the system completely. Secondly, for nonholonomic nonconservative systems, the constrained Herglotz equations are established, and the equations are transformed into partially canonical contravariant algebraic form by using integral factor. It is proved that the constrained Herglotz equations have a Lie admissibility algebraic structure, and Poisson theory is established. If the nonholonomic nonconservative system realizes free motion, then the constrained Herglotz equations have a Lie algebraic structure, and Poisson's theory can still be fully applied to the system. In this paper, we analyze the algebraic structure of constrained Herglotz equations and demonstrate the application of the obtained Poisson theory with the three examples, i.e., a nonlinear equation system, a ball rolling on a rough horizontal plane under uniform and isotropic Rayleigh dissipative forces, and the Appell-Hamel problem under viscous damping.