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E-raamat: Introduction to Global Variational Geometry

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Introduction to Global Variational GeometrySuurem pilt
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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.
1 Jet Prolongations of Fibered Manifolds
1(34)
1.1 The Rank Theorem
1(5)
1.2 Fibered Manifolds
6(4)
1.3 The Contact of Differentiable Mappings
10(3)
1.4 Jet Prolongations of Fibered Manifolds
13(4)
1.5 The Horizontalization
17(3)
1.6 Jet Prolongations of Automorphisms of Fibered Manifolds
20(3)
1.7 Jet Prolongations of Vector Fields
23(12)
References
33(2)
2 Differential Forms on Jet Prolongations of Fibered Manifolds
35(50)
2.1 The Contact Ideal
35(8)
2.2 The Trace Decomposition
43(10)
2.3 The Horizontalization
53(5)
2.4 The Canonical Decomposition
58(9)
2.5 Contact Components and Geometric Operations
67(1)
2.6 Strongly Contact Forms
68(6)
2.7 Fibered Homotopy Operators on Jet Prolongations of Fibered Manifolds
74(11)
References
84(1)
3 Formal Divergence Equations
85(18)
3.1 Formal Divergence Equations
85(4)
3.2 Integrability of Formal Divergence Equations
89(4)
3.3 Projectable Extensions of Differential Forms
93(10)
Reference
101(2)
4 Variational Structures
103(66)
4.1 Variational Structures on Fibered Manifolds
104(4)
4.2 Variational Derivatives
108(4)
4.3 Lepage Forms
112(11)
4.4 Euler--Lagrange Forms
123(1)
4.5 Lepage Equivalents and the Euler--Lagrange Mapping
124(5)
4.6 The First Variation Formula
129(1)
4.7 Extremals
130(3)
4.8 Trivial Lagrangians
133(2)
4.9 Source Forms and the Vainberg--Tonti Lagrangians
135(11)
4.10 The Inverse Problem of the Calculus of Variations
146(11)
4.11 Local Variationality of Second-Order Source Forms
157(12)
References
166(3)
5 Invariant Variational Structures
169(18)
5.1 Invariant Differential Forms
170(2)
5.2 Invariant Lagrangians and Conservation Equations
172(5)
5.3 Invariant Euler--Lagrange Forms
177(1)
5.4 Symmetries of Extremals and Jacobi Vector Fields
178(9)
References
185(2)
6 Examples: Natural Lagrange Structures
187(14)
6.1 The Hilbert Variational Functional
188(6)
6.2 Natural Lagrange Structures
194(3)
6.3 Connections
197(4)
References
200(1)
7 Elementary Sheaf Theory
201(62)
7.1 Sheaf Spaces
201(6)
7.2 Abelian Sheaf Spaces
207(4)
7.3 Sections of Abelian Sheaf Spaces
211(2)
7.4 Abelian Presheaves
213(4)
7.5 Sheaf Spaces Associated with Abelian Presheaves
217(4)
7.6 Sheaves Associated with Abelian Presheaves
221(5)
7.7 Sequences of Abelian Groups, Complexes
226(12)
7.8 Exact Sequences of Abelian Sheaves
238(4)
7.9 Cohomology Groups of a Sheaf
242(8)
7.10 Sheaves over Paracompact Hausdorff Spaces
250(13)
References
261(2)
8 Variational Sequences
263(40)
8.1 The Contact Sequence
264(8)
8.2 The Variational Sequence
272(1)
8.3 Variational Projectors
273(15)
8.4 The Euler--Lagrange Morphisms
288(8)
8.5 Variationally Trivial Lagrangians
296(2)
8.6 Global Inverse Problem of the Calculus of Variations
298(5)
References
300(3)
Appendix: Analysis on Euclidean Spaces and Smooth Manifolds 303(38)
Bibliography 341(6)
Index 347
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