In this work we introduce a category $ \mathfrak{L D P}_{d} $ of discrete-time dynamical systems, that we call discrete Lagrange–D'Alembert–Poincaré systems, and study some of its elementary properties. Examples of objects of $ \mathfrak{L D P}_{d} $ are nonholonomic discrete mechanical systems as well as their lagrangian reductions and, also, discrete Lagrange-Poincaré systems. We also introduce a notion of symmetry group for objects of $ \mathfrak{L D P}_{d} $ and a process of reduction when symmetries are present. This reduction process extends the reduction process of discrete Lagrange–Poincaré systems as well as the one defined for nonholonomic discrete mechanical systems. In addition, we prove that, under some conditions, the two-stage reduction process (first by a closed and normal subgroup of the symmetry group and, then, by the residual symmetry group) produces a system that is isomorphic in $ \mathfrak{L D P}_{d} $ to the system obtained by a one-stage reduction by the full symmetry group.

中文翻译：

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非完整离散机械系统的Lagrangian分解

在这项工作中，我们介绍了离散动力系统的类别\\ mathfrak {LDP} _ {d} $，我们将其称为离散Lagrange–D'Alembert–Poincaré系统，并研究了其一些基本性质。\ mathfrak {LDP} _ {d} $的对象的示例是非完整离散机械系统及其拉格朗日缩减，以及离散Lagrange-Poincaré系统。我们还为$ \ mathfrak {LDP} _ {d} $的对象引入了对称组的概念，以及在存在对称性时的简化过程。这种减少过程扩展了离散拉格朗日-庞加莱系统的减少过程，以及为非完整离散机械系统定义的过程。此外，我们证明了在某些条件下的两阶段还原过程（首先是对称组的一个封闭且正常的子组，然后是