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Infinite Dimensional Lagrange–Dirac Mechanics with Boundary Conditions

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Geometric Science of Information (GSI 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14072))

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Abstract

The Lagrange–Dirac theory is extended to systems defined on the family of smooth functions on a manifold with boundary, which provides an instance of systems with a Fréchet space as a configuration space. To that end, we introduce the restricted cotangent bundle, a vector subbundle of the topological cotangent bundle which contains the partial derivatives of Lagrangian functions defined through a density. The main achievement of our proposal is that the Lagrange–Dirac equations on the restricted cotangent bundle properly account for the boundary value problem, i.e., the boundary conditions do not need to be imposed ad hoc, but they arise naturally from the Lagrange–Dirac formulation. After giving the main theoretical results, and showing how boundary forces can be naturally included in the Lagrange–Dirac formulation, we illustrate our framework with the dynamical equations of a vibrating membrane.

ARA is supported by Ministerio de Universidades, Spain, under an FPU grant and partially supported by Ministerio de Ciencia e Innovación, Spain, under grant PID2021-126124NB-I00. FGB is partially supported by CNCS UEFISCDI, project number PN-III-P4-ID-PCE-2020-2888. HY is partially supported by JSPS Grant-in-Aid for Scientific Research (22K03443), JST CREST (JPMJCR1914), Waseda University (SR 2022C-423), and the MEXT “Top Global University Project”, SEES.

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Authors and Affiliations

  1. Instituto de Ciencias Matemáticas (CSIC–UAM–UC3M–UCM), Calle Nicolás Cabrera 13–15, Madrid, Spain

    Álvaro Rodríguez Abella

  2. Laboratoire de Météorologie Dynamique École Normale Supérieure/CNRS, Paris, France

    François Gay–Balmaz

  3. School of Science and Engineering Waseda University, 3–4–1, Okubo, Shinjuku, Tokyo, Japan

    Hiroaki Yoshimura

Authors
  1. Álvaro Rodríguez Abella
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  2. François Gay–Balmaz
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  3. Hiroaki Yoshimura
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Corresponding author

Correspondence to Álvaro Rodríguez Abella .

Editor information

Editors and Affiliations

  1. Sony Computer Science Laboratories Inc., Tokyo, Japan

    Frank Nielsen

  2. THALES Land and Air Systems, Meudon, France

    Frédéric Barbaresco

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Rodríguez Abella, Á., Gay–Balmaz, F., Yoshimura, H. (2023). Infinite Dimensional Lagrange–Dirac Mechanics with Boundary Conditions. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14072. Springer, Cham. https://doi.org/10.1007/978-3-031-38299-4_22

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  • DOI: https://doi.org/10.1007/978-3-031-38299-4_22

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-38298-7

  • Online ISBN: 978-3-031-38299-4

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