# American Institute of Mathematical Sciences

December  2019, 11(4): 487-510. doi: 10.3934/jgm.2019024

## 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  April 2018 Revised  April 2019 Published  November 2019

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.

Citation: María Barbero Liñán, Hernán Cendra, Eduardo García Toraño, David Martín de Diego. Morse families and Dirac systems. Journal of Geometric Mechanics, 2019, 11 (4) : 487-510. doi: 10.3934/jgm.2019024
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##### References:
The maps $\Psi_1$, $\Psi_2$ and $\Psi_3$
$D_M$ and $D_{\omega_Q}$