The Inverse Problem of the Calculus of Variations

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

   • Zbyněk 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, Zbyněk Urban

   Pages 215-284

8. Back Matter

   Pages 285-289

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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

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

  Dmitry V. Zenkov

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