Morse families and Dirac systems

Liñán María Barbero; Cendra Hernán; Toraño Eduardo García; Diego David Martín de

doi:10.3934/jgm.2019024

Article Contents

2019, Volume 11, Issue 4: 487-510. Doi: 10.3934/jgm.2019024

This issue Previous Article Networks of coadjoint orbits: From geometric to statistical mechanics Next Article Variational integrators for anelastic and pseudo-incompressible flows

Morse families and Dirac systems

1.
Departamento de Matemática Aplicada, Universidad Politécnica de Madrid, Av. Juan de Herrera 4, 28040 Madrid, Spain
2.
Departamento de Matemática, Universidad Nacional del Sur, CONICET, Av. Alem 1253, 8000 Bahía Blanca, Argentina
3.
Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/Nicolás Cabrera 13-15, 28049 Madrid, Spain

Received: March 31, 2018

Revised: March 31, 2019

Published: November 2019

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Abstract

Dirac structures and Morse families are used to obtain a geometric formalism that unifies most of the scenarios in mechanics (constrained calculus, nonholonomic systems, optimal control theory, higher-order mechanics, etc.), as the examples in the paper show. This approach generalizes the previous results on Dirac structures associated with Lagrangian submanifolds. An integrability algorithm in the sense of Mendella, Marmo and Tulczyjew is described for the generalized Dirac dynamical systems under study to determine the set where the implicit differential equations have solutions.

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Mathematics Subject Classification: Primary: 37J05, 70H03; Secondary: 34A26.

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Figure 1. The maps $ \Psi_1 $, $ \Psi_2 $ and $ \Psi_3 $

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Figure 2. $ D_M $ and $ D_{\omega_Q} $

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