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.

本文分别提出了两种不同的构造离散方程的方法和相应的连续 Birkhoff 系统的物理定律。制定并比较了相应的数学方法和几何结构。Noether 对称性的确定方程是通过作用于差分方程的李点变换获得的。利用规范函数满足的结构方程，给出了系统的两类离散守恒量。可以基于应用于具有对称性的非完整系统的这两种方法来开发算法。结果，几何结构和诺特不变量在数值上得以保留。