Singular reduction of implicit Hamiltonian systems

By Guido Blankenstein, Tudor S. Ratiu. 2003

51 citations recorded by the source corpus.

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.

Subfields: Control and Systems Engineering, Mathematical Physics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/6c5dda22c2b358317d6382e36026326477ce2343

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In Control and Systems Engineering

1. Shengda Hu
2. Hiroaki Yoshimura
3. Jerrold E. Marsden
4. Romeo Ortega
5. J. Alexander Fax

In Mathematical Physics

1. Jerrold E. Marsden
2. Hiroaki Yoshimura
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In Numerical Analysis

1. Jerrold E. Marsden
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4. Sanjay Lall
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6. Sina Ober-Blöbaum
7. Petr Krysl
8. Alessandro Astolfi

Citations in the corpus

• Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems (2006, methodology): “In the field of electric circuits and networks, the idea of interconnected systems, also referred to as nonenergic systems (or multiports), has been often employed in modeling and analysis of the systems, as in the work of Duinker [1959, 1962]; Kron [1963]; Wyatt and Chua [1977]; Brayton [1971]. The”
• Hamiltonian Symmetries and Reduction in Generalized Geometry (2005, background): “tion of Dirac structures comes from the relation with variational principles, see for example [42, 43]. The reduction in singular cases for the Poisson and Dirac structures are also known, e.g. [50], [11], as well as the excellent book [51], where many more references may be found. Group valued mome”