Lagrangian reduction of discrete mechanical systems by stages
2016
In this work we introduce a category of discrete
Lagrange--Poincare systems $\mathfrak{L}\mathfrak{P}_d$ and study some of its
properties. In particular, we show that the discrete mechanical
systems and the discrete dynamical systems obtained by the
Lagrangian reduction of symmetric discrete mechanical systems are
objects in $\mathfrak{L}\mathfrak{P}_d$. We introduce a notion of symmetry group for
objects of $\mathfrak{L}\mathfrak{P}_d$ as well as a reduction procedure that is
closed in the category $\mathfrak{L}\mathfrak{P}_d$. Furthermore, under some conditions,
we show that the reduction in two steps (first by a closed normal
subgroup of the symmetry group and then by the residual symmetry
group) is isomorphic in $\mathfrak{L}\mathfrak{P}_d$ to the reduction by the full
symmetry group.
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