Symmetries and exact solutions of discrete nonconservative systems

Based on the property of the discrete model entirely inheriting the symmetry of the continuous system, we present a method to construct exact solutions with continuous groups of transformations in discrete nonconservative systems. The Noether’s identity of the discrete nonconservative system is obtained. The symmetric discrete Lagrangian and symmetric discrete nonconservative forces are defined for the system. Generalized quasi-extremal equations of discrete nonconservative systems are presented. Discrete conserved quantities are derived with symmetries associated with the continuous system. We have also found that the existence of the one-parameter symmetry group provides a reduction to a conserved quantity; but the existence of a two-parameter symmetry group makes it possible to obtain an exact solution for a discrete nonconservative system. Several examples are discussed to illustrate these results.