This book provides a comprehensive introduction to modern global variational theory on fibred spaces. It is based on differentiation and integration theory of differential forms on smooth manifolds, and on the concepts of global analysis and geometry such as jet prolongations of manifolds, mappings, and Lie groups.

The book will be invaluable for researchers and PhD students in differential geometry, global analysis, differential equations on manifolds, and mathematical physics, and for the readers who wish to undertake further rigorous study in this broad interdisciplinary field.

Featured topics

- Analysis on manifolds

- Differential forms on jet spaces

- Global variational functionals

- Euler-Lagrange mapping

- Helmholtz form and the inverse problem

- Symmetries and the Noether??'s theory of conservation laws

- Regularity and the Hamilton theory

- Variational sequences

- Differential invariants and natural variational principles

- First book on the geometric foundations of Lagrange structures

- New ideas on global variational functionals

- Complete proofs of all theorems

- Exact treatment of variational principles in field theory, inc. general relativity

- Basic structures and tools: global analysis, smooth manifolds, fibred spaces

Since the very beginning of a period of great expansion of the theory of smooth manifolds, about 1960th, the rapid developments in this field have substantially influenced many branches of modern global analysis, geometry, and mathematical physics. The book is intended to explain the developments of the global variational theory on fibered spaces in last decades. Our principal aim is to present a fundamental, complete work in this field of modern mathematics. We also aim to give an exact exposition of applications of this theory in mathematical and theoretical physics.

- Presents a complete exposition of basic concepts as well as complete proofs of all basic results in this theory

- Characterizes the contributions of contemporary researchers and founders of this modern mathematical discipline, to cover and evaluate essential sources and references (bibliography included)

- Formulate the theory in a closed manner, and to make the proofs self-contained (necessary basic results of algebra, topology, and analysis are included)

Preface

List of Standard Symbols

Chapter 1: Smooth Manifolds

Chapter 2: Analysis on Manifolds

Chapter 3: Lie Transformation Groups

Chapter 4: Lagrange Structures

Chapter 5: Elementary Sheaf Theory

Chapter 6: Variational Sequences on Fibered Manifolds

Chapter 7: Invariant Variational Functionals on Principal Bundles

Chapter 8: Differential Invariants

Chapter 9: Natural Variational Principles

Appendices

Bibliography

Index