The Inverse Problem of the Calculus of Variations

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).

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

Contenu
The Helmholtz Conditions and the Method of Controlled Lagrangians.- The SoninDouglas Problem.- Inverse Variational Problem and Symmetry in Action: The Relativistic Third Order Dynamics.- Variational Principles for Immersed Submanifolds.- Source Forms and their Variational Completions.- First-Order Variational Sequences in Field Theory.