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First variation formula for discrete variational problems in two independent variables

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Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

We introduce a geometric setup for discrete variational problems in two independent variables based on the theory of bundles modelled on cell complexes, and characterize geometrically a first variation formula and a discrete Noether theorem for symmetries of the discrete Lagrangian. We explore the existence of discrete variational integrators, which will conserve the discrete Noether currents in the theory. The range of applications of the theory includes discrete versions of variational problems with or without PDE constraints and energy-conserving integrators in mechanics of continua. We illustrate the theory with a variational integrator for the movement of a string.

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Authors and Affiliations

  1. Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516, Caparica, Portugal

    Ana Cristina Casimiro

  2. Departamento de Matemática, Instituto Superior Técnico, CAMGSD, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Ana Cristina Casimiro

  3. CINAMIL, Academia Militar, Avda. Conde Castro Guimaraes, 2720-113, Amadora, Portugal

    César Rodrigo

  4. CMAF, Centro de Matemática e Aplicações Fundamentais, Lisboa, Portugal

    César Rodrigo

Authors
  1. Ana Cristina Casimiro
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  2. César Rodrigo
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Corresponding author

Correspondence to Ana Cristina Casimiro.

Additional information

A. C. Casimiro has been partially supported by Junta de Castilla y León (Spain), Project SA112A07. C. Rodrigo has been partially supported by Plan Nacional de I+D+i, Ministerio de Educación y Ciencia (Spain), Project MTM2007-60017.

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Casimiro, A.C., Rodrigo, C. First variation formula for discrete variational problems in two independent variables. RACSAM 106, 111–135 (2012). https://doi.org/10.1007/s13398-011-0034-6

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  • DOI: https://doi.org/10.1007/s13398-011-0034-6

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