This paper is devoted to discrete mechanical systems subject to external forces. We introduce a discrete version of systems with Rayleigh-type forces, obtain the equations of motion and characterize the equivalence for these systems. Additionally, we obtain a Noether's theorem and other theorem characterizing the Lie subalgebra of symmetries of a forced discrete Lagrangian system. Moreover, we develop a Hamilton-Jacobi theory for forced discrete Hamiltonian systems. These results are useful for the construction of so-called variational integrators, which, as we illustrate with some examples, are remarkably superior to the usual numerical integrators such as the Runge-Kutta method.

翻译：本文专门论述受外部力量制约的离散机械系统。 我们用雷利格型的力量引入离散的系统版本, 获得运动方程式, 并描述这些系统的等同性。 此外, 我们获得了诺埃瑟人的理论和其他理论, 它们是强制离散的拉格朗加系统对称的精细亚代数。 此外, 我们为被迫离散的汉密尔顿系统开发了汉密尔顿- 贾科比理论。 这些结果对于建造所谓的变异集成器很有用, 正如我们用一些例子所说明的那样, 与诸如龙格- 库塔方法等通常的数字集成器相比, 特别优越。