Singular reduction of implicit Hamiltonian systems |
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Authors: | Guido Blankenstein |
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Institution: | Department of Mechanical Engineering, Katholieke Universiteit Leuven, Celestijnenlaan 300 B, B-3001 Leuven (Heverlee), Belgium; Centre Bernoulli, École Polytechnique Fédérale de Lausanne, MA-Ecublens, CH-1015 Lausanne, Switzerland |
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Abstract: | This paper develops the theory of singular reduction for implicit Hamiltonian systems admitting a symmetry Lie group. The reduction is performed at a singular value of the momentum map. This leads to a singular reduced topological space which is not a smooth manifold. A topological Dirac structure on this space is defined in terms of a generalized Poisson bracket and a vector space of derivations, both being defined on a set of smooth functions. A corresponding Hamiltonian formalism is described. It is shown that solutions of the original system descend to solutions of the reduced system. Finally, if the generalized Poisson bracket is nondegenerate, then the singular reduced space can be decomposed into a set of smooth manifolds called pieces. The singular reduced system restricts to a regular reduced implicit Hamiltonian system on each of these pieces. The results in this paper naturally extend the singular reduction theory as previously developed for symplectic or Poisson Hamiltonian systems. |
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Keywords: | implicit Hamiltonian systems Dirac structures symmetry reduction |
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