Two forms of the discrete equations and the Noether theorems for nonautonomous Birkhoffian systems

Two different ways of constructing the discrete equations and the corresponding physical laws of continuous Birkhoffian systems are respectively proposed in this paper. The corresponding mathematical methods and geometric structures are formulated and compared. The determining equations of the Noether symmetries are obtained via the Lie point transformations acting on the difference equations. Two types of the discrete conserved quantities of the systems are presented using the structure equation satisfied by the gauge functions. The algorithms can be developed based on these two approaches applied to the nonholonomic systems with symmetries. As a result, the geometric structure and the Noether invariants are numerically preserved. The numerical simulations based on the two approaches demonstrate the high precision and the long-time stability of the algorithms compared with the standard Runge-Kutta method.