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The Inverse Problem of the Calculus of Variations

Local and Global Theory

  • Book
  • © 2015

Overview

Editors:
  1. Dmitry V. Zenkov

    1. North Carolina State University Dept. Mathematics, Raleigh, USA

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  • Unified exposition of the inverse variational problem for ordinary and partial differential equations and for equations on manifolds
  • First systematic contribution to the global inverse problem of the calculus of variations based on modern differential geometry and algebraic topology
  • Selected applications of the inverse problem in geometry, optimal control theory and modern theoretical physics (higher-order mechanics and general relativity)
  • Prepares the reader for research in the local and global inverse problem using variational sequence theory and its consequences based on elementary sheaf theory
  • Includes supplementary material: sn.pub/extras

Part of the book series: Atlantis Studies in Variational Geometry (ASVG, volume 2)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-ix
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  2. The Helmholtz Conditions and the Method of Controlled Lagrangians

    • Anthony M. Bloch, Demeter Krupka, Dmitry V. Zenkov
    Pages 1-29

  3. The Sonin–Douglas Problem

    • Demeter Krupka
    Pages 31-73

  4. Inverse Variational Problem and Symmetry in Action: The Relativistic Third Order Dynamics

    • Roman Ya. Matsyuk
    Pages 75-102

  5. Variational Principles for Immersed Submanifolds

    • ZbynÄ›k Urban
    Pages 103-170

  6. Source Forms and Their Variational Completions

    • Nicoleta Voicu
    Pages 171-214

  7. First-Order Variational Sequences in Field Theory

    • Jana Volná, ZbynÄ›k Urban
    Pages 215-284

  8. Back Matter

    Pages 285-289
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Keywords

About this book

The aim of the present book is to give a systematic treatment of the inverse problem of the calculus of variations, i.e. how to recognize whether a system of differential equations can be treated as a system for extremals of a variational functional (the Euler-Lagrange equations), using contemporary geometric methods. Selected applications in geometry, physics, optimal control, and general relativity are also considered. The book includes the following chapters: - Helmholtz conditions and the method of controlled Lagrangians (Bloch, Krupka, Zenkov) - The Sonin-Douglas's problem (Krupka) - Inverse variational problem and symmetry in action: The Ostrogradskyj relativistic third order dynamics (Matsyuk.) - Source forms and their variational completion (Voicu) - First-order variational sequences and the inverse problem of the calculus of variations (Urban, Volna) - The inverse problem of the calculus of variations on Grassmann fibrations (Urban).

Editors and Affiliations

  • North Carolina State University Dept. Mathematics, Raleigh, USA

    Dmitry V. Zenkov

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