论文标题
通过阶段减少非体力学离散机械系统的拉格朗日
Lagrangian reduction of nonholonomic discrete mechanical systems by stages
论文作者
论文摘要
在这项工作中，我们介绍了离散时间动态系统的类别$ LDP_D $，我们称之为离散的Lagrange--d'Alembert--Poincaré系统，并研究其一些基本属性。 $ LDP_D $的对象的示例是非独立的离散机械系统及其Lagrangian减少以及离散的Lagrange-Poincaré系统。我们还为$ LDP_D $的对象引入了对称组的概念，并在存在对称性时进行了减少。这种还原过程扩展了离散Lagrange-Poincaré系统的还原过程，以及针对非机学离散机械系统定义的过程。此外，我们证明，在某些条件下，两阶段的减少过程（首先是由对称组的封闭式和正常亚组，然后由残留对称组的封闭和正常亚组产生的系统，该系统是$ LDP_D $ in $ LDP_D $的系统，由完整对称组的一阶段减少的系统获得。
In this work we introduce a category $LDP_d$ 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 $LDP_d$ 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 $LDP_d$ 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 $LDP_d$ to the system obtained by a one-stage reduction by the full symmetry group.
