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Threedimensional discrete systems of HirotaKimura type and deformed LiePoisson algebras
Dirac cotangent bundle reduction
1.  Applied Mechanics and Aerospace Engineering, Waseda University, Okubo, Shinjuku, Tokyo 1698555, Japan 
2.  Control and Dynamical Systems 10781, California Institute of Technology, Pasadena, CA 91125, United States 
First of all, we establish a reduction theory starting with the HamiltonPontryagin variational principle, which enables one to formulate an implicit analogue of the LagrangePoincaré equations. To do this, we assume that a Lie group acts freely and properly on a configuration manifold, in which case there is an associated principal bundle and we choose a principal connection. Then, we develop a reduction theory for the canonical Dirac structure on the cotangent bundle to induce a gauged Dirac structure . Second, it is shown that by making use of the gauged Dirac structure, one obtains a reduction procedure for standard implicit Lagrangian systems, which is called LagrangePoincaréDirac reduction . This procedure naturally induces the horizontal and vertical implicit LagrangePoincaré equations , which are consistent with those derived from the reduced HamiltonPontryagin principle. Further, we develop the case in which a Hamiltonian is given (perhaps, but not necessarily, coming from a regular Lagrangian); namely, HamiltonPoincaréDirac reduction for the horizontal and vertical HamiltonPoincaré equations . We illustrate the reduction procedures by an example of a satellite with a rotor.
The present work is done in a way that is consistent with, and may be viewed as a specialization of the larger context of Dirac reduction, which allows for Dirac reduction by stages . This is explored in a paper in preparation by Cendra, Marsden, Ratiu and Yoshimura.
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