An extension of the Dirac and Gotay-Nester theories of constraints for Dirac dynamical systems

Hernán Cendra; María Etchechoury; Sebastián J. Ferraro

doi:10.3934/jgm.2014.6.167

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June 2014, 6(2): 167-236. doi: 10.3934/jgm.2014.6.167

An extension of the Dirac and Gotay-Nester theories of constraints for Dirac dynamical systems

Hernán Cendra ^1, , María Etchechoury ^2, and Sebastián J. Ferraro ^3,

1.	Departamento de Matemática, Universidad Nacional del Sur, Av. Alem 1253, 8000 Bahía Blanca and CONICET
2.	Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, CC 172, 1900 La Plata, Argentina
3.	Departamento de Mateemática and Instituto de Matemática Bahía Blanca, Universidad Nacional del Sur, Av. Alem 1253, 8000 Bahía Blanca and CONICET, Argentina

Received March 2011 Revised May 2014 Published June 2014

Abstract
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This paper extends the Gotay-Nester and the Dirac theories of constrained systems in order to deal with Dirac dynamical systems in the integrable case. Integrable Dirac dynamical systems are viewed as constrained systems where the constraint submanifolds are foliated. The cases considered usually in the literature correspond to a trivial foliation, with only one leaf. A Constraint Algorithm for Dirac dynamical systems (CAD), which extends the Gotay-Nester algorithm, is developed. Evolution equations are written using a Dirac bracket adapted to the foliations and an adapted total energy. The interesting example of LC circuits is developed in detail. The paper emphasizes the point of view that Dirac and Gotay-Nester theories are, in a certain sense, dual and that using a combination of results from both theories may have advantages in dealing with a given example, rather than using systematically one or the other.

Keywords: Constrained systems, Dirac structures, nonholonomic mechanics, Poisson geometry., implicit differential equations.

Mathematics Subject Classification: Primary: 70H45, 70G45, 70H03, 70H05; Secondary: 34A0.

Citation: Hernán Cendra, María Etchechoury, Sebastián J. Ferraro. An extension of the Dirac and Gotay-Nester theories of constraints for Dirac dynamical systems. Journal of Geometric Mechanics, 2014, 6 (2) : 167-236. doi: 10.3934/jgm.2014.6.167

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