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Lagrangian reduction of nonholonomic discrete mechanical systems by stages

This research was partially supported by grants from Universidad Nacional de Cuyo (#06/C567 and #06/C574) and Universidad Nacional de La Plata

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  • 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.

    Mathematics Subject Classification: Primary: 37J06, 70G45; Secondary: 70G75.


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