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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.

### Chapter 1. Jet Prolongations of Fibered Manifolds

Abstract
This chapter introduces fibered manifolds and their jet prolongations. First, we recall properties of differentiable mappings of constant rank and introduce, with the help of rank, the notion of a fibered manifold. Then, we define automorphisms of fibered manifolds as the mappings preserving their fibered structure. The r-jets of sections of a fibered manifold Y, with a fixed positive integer r, constitute a new fibered manifold, the r-jet prolongation $$J^{r} Y$$ of Y; we describe the structure of $$J^{r} Y$$ and a canonical construction of automorphisms of $$J^{r} Y$$ from automorphisms of the fibered manifold Y, their r-jet prolongation. The prolongation procedure immediately extends, via flows, to vector fields.
Demeter Krupka

### Chapter 2. Differential Forms on Jet Prolongations of Fibered Manifolds

Abstract
In this chapter, we present a decomposition theory of differential forms on jet prolongations of fibered manifolds; the tools inducing the decompositions are the algebraic trace decomposition theory and the canonical jet projections. Of particular interest is the structure of the contact forms, annihilating integrable sections of the jet prolongations. We also study decompositions of forms defined by fibered homotopy operators and state the corresponding fibered Poincare-Volterra lemma.
Demeter Krupka

### Chapter 3. Formal Divergence Equations

Abstract
In this chapter, we introduce formal divergence equations on Euclidean spaces and study their basic properties. These partial differential equations naturally appear in the variational geometry on fibered manifolds, but also have a broader meaning related to differential equations, conservation laws, and integration of forms on manifolds with boundary. A formal divergence equation is not always integrable; we show that the obstructions are connected with the EulerLagrange expressions known from the higher-order variational theory of multiple integrals. If a solution exists, then it defines a solution of the associated “ordinary” divergence equation along any section of the underlying fibered manifold. The notable fact is that the solutions of formal divergence equations of order r are in one–one correspondence with a class of differential forms on the $$\boldsymbol{(r - 1)}$$-st jet prolongation of the underlying fibered manifold, defined by the exterior derivative operator.
Demeter Krupka

### Chapter 4. Variational Structures

Abstract
In this chapter, a complete treatment of the foundations of the calculus of variations on fibered manifolds is presented. Using the calculus of differential forms as the main tool, the aim is to study higher-order integral variational functionals of the orm $$\gamma \to {\int }J^{r} \gamma \ast \rho$$, depending on sections γ of a fibered manifold Y, where ρ is a general differential form on the jet manifold $$J^{r} Y$$ and $$J^{r} \gamma$$ is the r-jet prolongation γ. The horizontal forms ρ are the Lagrangians. Variations (deformations) of sections of Y are considered as vector fields, permuting the set of sections, and their prolongations to the jet manifolds $$J^{r} Y$$. They are applied to the variational functionals in a geometric way by means of the Lepage forms. The main idea can be introduced by means of the Cartan’s formula for the Lie derivative of a differential form η on a manifold Z, $${\mathrm{\partial}}_{\xi } \eta = i_{\xi } {{d}}\eta + {{d}}i_{\xi } \eta$$, where $$i_{\xi }$$ is the contraction by a vector field ξ and d is the exterior derivative operator. Replacing Z with the r-jet prolongation $$J^{r} Y$$ and η with ρ, it is proved that the form ρ in the variational functional $$\gamma \to {\int }J^{r} \gamma \ast \rho$$ may be chosen in such a way that the Cartan’s formula forρ becomes a geometric version of the classical first variation formula. A structure theorem implies that for different underlying manifold structures and order of their jet prolongations, the concept of a Lepage formgeneralizes the well-known Cartan form in classical mechanics, the Poincaré-Cartan forms in the first-order field theory, the so-called fundamental forms, the second-order generalisation of the Poincaré-Cartan form, the Carathéodory form, and the Hilbert form in Finsler geometry.
Demeter Krupka

### Chapter 5. Invariant Variational Structures

Abstract
Let X be any manifold, W an open set in X, and let α: W → X be a smooth mapping. A differential form η, defined on the set α(W) in X, is said to be invariant with respect to α, if the transformed form $$\alpha \ast \eta$$ coincides with η, that is, if $$\alpha \ast \eta = \eta$$ on the set $$W \cap \alpha (W)$$; in this case, we also say that α is an invariance transformation of η. A vector field, whose local one-parameter group consists of invariance transformations of η, is called the generator of invariance transformations. In this chapter, these definitions are extended to variational structures (Y, ρ) and to the integral variational funtionals associated with them. Our objective is to study invariance properties of the form ρ and other differential forms, associated with ρ, the Lagrangian $$\lambda$$, and the Euler-Lagrange from $$E_{\lambda }$$. The class of transformations we consider are auto-morphisms of fibred manifolds and their jet prolongations. This part of the variational theory represents a notable extension of the classical coordinate concepts and methods to topologically nontrivial fibred manifolds that cannot be covered by a single chart. The geometric coordinate-free structure of the infinitesimal first variation formula leads in several consequences, such as the geometric invariance criteria of the Lagrangians and the Euler-Lagrange forms, a global theorem on the conservation law equations, and the relationship between extremals and conservation laws. Resuming, we can say that these results as a whole represent an extension of the classical Noether’s theory to higher-order variational functional on fibred manifolds
Demeter Krupka

### Chapter 6. Examples: Natural Lagrange Structures

Abstract
Examples presented in this chapter include typical variational functionals that appear as variational principles in the theory of geometric and physical fields. We begin by the discussion of the well-known Hilbert variational functional for the metric fields, first considered in Hilbert in 1915, whose Euler–Lagrange equations are the Einstein vacuum equations. We give a manifold interpretation of this functional and show that its second-order Lagrangian, the formal scalar curvature, possesses a global first-order Lepage equivalent. The Lagrangian used by Hilbert is an example of a differential invariant of a metric field (and its first and second derivatives). Further examples with similar properties, belonging to the class of natural Lagrange structures, are also considered.
Demeter Krupka

### Chapter 7. Elementary Sheaf Theory

Abstract
The purpose of this chapter is to explain selected topics of the sheaf theory over paracompact, Hausdorff topological spaces. The choice of questions we consider are predetermined by the global variational theory over (topologically nontrivial) fibered manifolds, namely by the problem how to characterize differences between the local and global properties of the Euler–Lagrange mapping, between locally and globally trivial Lagrangians, and locally and globally variational source forms. To this purpose, the central topic we follow is the abstract De Rham theorem and its consequences. In particular, in the context of this book, the cohomology of abstract sheaves should be compared with the cohomology of the associated complexes of global sections, and the cohomology of underlying smooth manifolds.
Demeter Krupka

### Chapter 8. Variational Sequences

Abstract
We introduced in Chap. 4 the Euler–Lagrange mapping of the calculus of variations as an $${\mathbf{R}}$$-linear mapping, assigning to a Lagrangian $$\lambda$$, defined on the r-jet prolongation $$J^{r} Y$$ of a fibered manifold Y, its Euler–Lagrange form $$E_{\lambda }$$. Local properties of this mapping are determined by the components of the Euler–Lagrange form, the Euler–Lagrange expressions of the Lagrangian $$\lambda$$. In this chapter, we construct an exact sequence of Abelian sheaves, the variational sequence, such that one of its sheaf morphisms coincides with the Euler–Lagrange mapping. Existence of the sequence provides a possibility to study basic global characteristics of the Euler–Lagrange mapping in terms of the cohomology groups of the corresponding complex of global sections and the underlying manifold Y. In particular, for variational purposes, the structure of the kernel and the image of the Euler–Lagrange mapping $$\lambda \to E_{\lambda }$$ is considered.
Demeter Krupka