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Introduction to Global Variational Geometry

  • Book
  • © 2015

Overview

Authors:
  1. Demeter Krupka

    1. Department of Mathematics and Lepage Research Institute, University of Hradec Kralove, Hradec Kralove, Czech Republic

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  • Systematic exposition of the higher-order global variational theory on fibered spaces with complete proofs of basic results
  • First systematic treatment of variational calculus and its applications for manifolds
  • Covers applications of variational theory in modern theoretical physics
  • Prepares the reader for research in the local and global inverse problem of the calculus of variations for partial differential equations
  • Variational sequence theory and its consequences are explained, based on elementary sheaf theory
  • Includes supplementary material: sn.pub/extras

Part of the book series: Atlantis Studies in Variational Geometry (ASVG, volume 1)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xvii
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  2. Jet Prolongations of Fibered Manifolds

    • Demeter Krupka
    Pages 1-33

  3. Differential Forms on Jet Prolongations of Fibered Manifolds

    • Demeter Krupka
    Pages 35-84

  4. Formal Divergence Equations

    • Demeter Krupka
    Pages 85-101

  5. Variational Structures

    • Demeter Krupka
    Pages 103-168

  6. Invariant Variational Structures

    • Demeter Krupka
    Pages 169-185

  7. Examples: Natural Lagrange Structures

    • Demeter Krupka
    Pages 187-200

  8. Elementary Sheaf Theory

    • Demeter Krupka
    Pages 201-261

  9. Variational Sequences

    • Demeter Krupka
    Pages 263-301

  10. Back Matter

    Pages 303-354
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Keywords

About this book

The book is devoted to recent research in the global variational theory on smooth manifolds. Its main objective is an extension of the classical variational calculus on Euclidean spaces to (topologically nontrivial) finite-dimensional smooth manifolds; to this purpose the methods of global analysis of differential forms are used. Emphasis is placed on the foundations of the theory of variational functionals on fibered manifolds - relevant geometric structures for variational principles in geometry, physical field theory and higher-order fibered mechanics. The book chapters include: - foundations of jet bundles and analysis of differential forms and vector fields on jet bundles, - the theory of higher-order integral variational functionals for sections of a fibred space, the (global) first variational formula in infinitesimal and integral forms- extremal conditions and the discussion of Noether symmetries and generalizations,- the inverse problems of the calculus of variations of Helmholtz type- variational sequence theory and its consequences for the global inverse problem (cohomology conditions)- examples of variational functionals of mathematical physics. Complete formulations and proofs of all basic assertions are given, based on theorems of global analysis explained in the Appendix.

Authors and Affiliations

  • Department of Mathematics and Lepage Research Institute, University of Hradec Kralove, Hradec Kralove, Czech Republic

    Demeter Krupka

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