Discrete dynamics  in implicit form

David Iglesias-Ponte; Juan Carlos Marrero; David Martín de Diego; Edith Padrón

doi:10.3934/dcds.2013.33.1117

Article Contents

2013, Volume 33, Issue 3: 1117-1135. Doi: 10.3934/dcds.2013.33.1117

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Discrete dynamics in implicit form

1.
Unidad asociada ULL-CSIC “Geometría Diferencial y Mecánica Geométrica”, Departamento de Matemática Fundamental, Facultad de Matemáticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands
2.
Unidad asociada ULL-CSIC Geometría Diferencial y Mecánica Geométrica, Departamento de Matemática Fundamental, Facultad de Matemáticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands
3.
Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Campus de Cantoblanco, UAM, C/Nicolás Cabrera, 15, 28049 Madrid

Received: March 31, 2011

Revised: October 31, 2011

Published: February 2013

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Abstract

A notion of implicit difference equation on a Lie groupoid is introduced and an algorithm for extracting the integrable part (backward or/and forward) is formulated. As an application, we prove that discrete Lagrangian dynamics on a Lie groupoid $G$ may be described in terms of Lagrangian implicit difference equations of the corresponding cotangent groupoid $T^*G$. Other situations include finite difference methods for time-dependent linear differential-algebraic equations and discrete nonholonomic Lagrangian systems, as parti-cular examples.

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Mathematics Subject Classification: Primary: 34K32, 22A22; Secondary: 17B66, 37M15, 57D17.

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