© 2015

The Inverse Problem of the Calculus of Variations

Local and Global Theory

  • Dmitry V. Zenkov

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

Table of contents

  1. Front Matter
    Pages i-ix
  2. Anthony M. Bloch, Demeter Krupka, Dmitry V. Zenkov
    Pages 1-29
  3. Demeter Krupka
    Pages 31-73
  4. Nicoleta Voicu
    Pages 171-214
  5. Jana Volná, Zbyněk Urban
    Pages 215-284
  6. Back Matter
    Pages 285-289

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).


Euler-Lagrange form Helmholtz conditions Lagrangian Source form Variational sequence

Editors and affiliations

  • Dmitry V. Zenkov
    • 1
  1. 1.North Carolina State University Dept. MathematicsRaleighUSA

Bibliographic information