Introduction to Global Variational Geometry

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.

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

Contenu
Jet prolongations of fibred manifolds.- Differential forms on jet prolongations of fibred manifolds.- Formal divergence equations.- Variational structures.- Invariant variational structures.- Examples: Natural Lagrange structures.- Elementary sheaf theory.- Variational sequences.