Introduction to global variational geometry.

*(English)*Zbl 1310.49001
Atlantis Studies in Variational Geometry 1. Amsterdam: Atlantis Press (ISBN 978-94-6239-072-0/hbk; 978-94-6239-073-7/ebook). xvii, 354 p. (2015).

The author of this book is a well known specialist in the theory of variational principles on jet bundles, natural bundles and differential invariants, differential equations and geometric structures. His main research areas are global variational analysis, differential geometry, tensor algebra and mathematical physics. Among his most known achievments we may quote: development of a global theory of functionals in fibred spaces, the higher order generalizations of this theory based on the concept of Lepage form and the Euler-Lagrange form. He participated actively in editing the Proceeedings of 11 editions of the Conference Differential Geometry and Appplications and is an active editor of 7 mathematical journals of very good quality. Let us remark that he is a founding editor of the well known journal Differential Geometry and its Appplications. D. Krupka is an author of several textbooks, translations and several survey papers and chapters in some proceedings and handbooks.

The subjects from the present book are considered as basic in the works on the (local) calculus of variations on Euclidean spaces: variational functionals and their variations, the first and second variation formula, extremals and the Euler-Lagrange equations, invariance and conservation laws. In the present book, the author studies these topics under much broader underlying structures, smooth manifolds. The problem of the global existence of some notions constructed in local charts is considered. E.g., the global properties of Euler-Lagrange mappings are studied. Modern geometric methods in the calculus of variations on manifolds are used, too, which are presented in several recent monographs.

In chapter 1, the fundamentals of fibered manifolds and their jet prolongations are studied. After presenting the usual topics about the jet structure one studies the horizontalization morphism, jet prolongations of sections and morphisms of fibered manifolds as well as the prolongations of vector fields. In the second chapter, one studies the differential forms on jet prolongations of fibered manifolds. The contact forms are introduced, generating a differential ideal of the exterior algebra. Concerning this concept one studies the corresponding decompositions of forms. The trace operation acting on the components of forms leads to a decomposition related to the exterior derivative of forms. In chapter 3, the formal divergence equations on jet manifolds are presented. The integrability of these equations is equivalent to the vanishing of the Euler-Lagrange operator. The fundamental notion studied in the present book is that of variational structure. It is a pair \((Y,\rho)\), where \(Y\) is a fibered manifold over an \(n\)-dimensional manifold \(X\) with projection \(\pi\) and \(\rho\) is an \(n\)-form on the \(r\)-jet prolongation \(J^rY\). In chapters 4-6, one studies the behavior of the variational functional \(\rho_\Omega (\gamma)=\int J^r\gamma^*\rho\) with respect to the sections \(\gamma\) of the fibered manifold \(Y\). The following key notions are introduced and studied: the variational derivative, Lepage form, the first variation formula, Euler-Lagrange form, trivial Lagrangian, source form and Vainberg-Tonti Lagrangian. Then, the inverse problem of the calculus of variations and the Helmholtz expressions are studied. For any variational structure \((Y,\rho)\) there exists the Lepage \(n\)-form \(\Theta_\rho\) which defines the same integral variational functional as the form \(\rho\) and the exterior differential \(d\Theta_\rho\) defines equations for the extremals for the sections \(\gamma\) of the fibered manifold \(Y\). The exterior differential \(d\Theta_\rho\) splits in two terms, one of them characterizing extremals. This is the Euler-Lagrange form, leading to the Euler-Lagrange equations for the sections in \(Y\). A careful analysis of the structure of the Euler-Lagrange mappings (assigning to every Lagrangian its Euler-Lagrange form) is made. The author presents a description of the kernel and the image of the Euler-Lagrange mapping. Then, he studies the variational structures whose Lagrangians or Euler-Lagrange forms admit some invariance transformations. One finds a generalization of Noether’s theorem for a given variational structure \((Y,\rho)\) relating the generators of invariance transformations of \(\rho\) with the existence of conservation laws for the solutions of the system of Euler-Lagrange equations. Next, the author considers a few examples of natural variational structures in order to establish basic structures and to find the corresponding Lepage forms. One discusses the Hilbert variational functional for the metric field and a variational functional for connections. By using the Euler-Lagrange mapping which assigns to an \(n\)-form \(\lambda\) (the Lagrangian) an \((n+1)\)-form \(E_\lambda\) (the Euler-Lagrange form) by using the exterior differential operator \(d\), an appropriate canonical decomposition of the underlying spaces of forms and the concept of a Lepage form, one can include the Euler-Lagrange mapping in a differential sequence of sheaves. It follows the possibility to study the global properties of the Euler-Lagrange mapping. After explaining some elements of the sheaf theory, insisting on those which are important for the variational structures, the author studies the critical points of the variational functionals and some geometric problems connected to the various kinds of symmetries of the variational functionals. Several variational principles in physical field theory and geometric mechanics are obtained as particular cases of the general theory. The last chapter (8) is devoted to the study of the variational sequence of order \(r\) for a fibered manifold \(Y\). Its construction is based on the fact that the de Rham sequence of differential forms on the \(r\)-jet prolongation \(J^rY\) has a remarkable subsequence defined by the contact forms. The variational sequence is obtained as the quotient sheaf of de Rham sequence. The author considers some local properties of the terms of the quotient sequence and global properties represented by theorems on the cohomology of the complex of global sections of the variational sequence.

The subjects from the present book are considered as basic in the works on the (local) calculus of variations on Euclidean spaces: variational functionals and their variations, the first and second variation formula, extremals and the Euler-Lagrange equations, invariance and conservation laws. In the present book, the author studies these topics under much broader underlying structures, smooth manifolds. The problem of the global existence of some notions constructed in local charts is considered. E.g., the global properties of Euler-Lagrange mappings are studied. Modern geometric methods in the calculus of variations on manifolds are used, too, which are presented in several recent monographs.

In chapter 1, the fundamentals of fibered manifolds and their jet prolongations are studied. After presenting the usual topics about the jet structure one studies the horizontalization morphism, jet prolongations of sections and morphisms of fibered manifolds as well as the prolongations of vector fields. In the second chapter, one studies the differential forms on jet prolongations of fibered manifolds. The contact forms are introduced, generating a differential ideal of the exterior algebra. Concerning this concept one studies the corresponding decompositions of forms. The trace operation acting on the components of forms leads to a decomposition related to the exterior derivative of forms. In chapter 3, the formal divergence equations on jet manifolds are presented. The integrability of these equations is equivalent to the vanishing of the Euler-Lagrange operator. The fundamental notion studied in the present book is that of variational structure. It is a pair \((Y,\rho)\), where \(Y\) is a fibered manifold over an \(n\)-dimensional manifold \(X\) with projection \(\pi\) and \(\rho\) is an \(n\)-form on the \(r\)-jet prolongation \(J^rY\). In chapters 4-6, one studies the behavior of the variational functional \(\rho_\Omega (\gamma)=\int J^r\gamma^*\rho\) with respect to the sections \(\gamma\) of the fibered manifold \(Y\). The following key notions are introduced and studied: the variational derivative, Lepage form, the first variation formula, Euler-Lagrange form, trivial Lagrangian, source form and Vainberg-Tonti Lagrangian. Then, the inverse problem of the calculus of variations and the Helmholtz expressions are studied. For any variational structure \((Y,\rho)\) there exists the Lepage \(n\)-form \(\Theta_\rho\) which defines the same integral variational functional as the form \(\rho\) and the exterior differential \(d\Theta_\rho\) defines equations for the extremals for the sections \(\gamma\) of the fibered manifold \(Y\). The exterior differential \(d\Theta_\rho\) splits in two terms, one of them characterizing extremals. This is the Euler-Lagrange form, leading to the Euler-Lagrange equations for the sections in \(Y\). A careful analysis of the structure of the Euler-Lagrange mappings (assigning to every Lagrangian its Euler-Lagrange form) is made. The author presents a description of the kernel and the image of the Euler-Lagrange mapping. Then, he studies the variational structures whose Lagrangians or Euler-Lagrange forms admit some invariance transformations. One finds a generalization of Noether’s theorem for a given variational structure \((Y,\rho)\) relating the generators of invariance transformations of \(\rho\) with the existence of conservation laws for the solutions of the system of Euler-Lagrange equations. Next, the author considers a few examples of natural variational structures in order to establish basic structures and to find the corresponding Lepage forms. One discusses the Hilbert variational functional for the metric field and a variational functional for connections. By using the Euler-Lagrange mapping which assigns to an \(n\)-form \(\lambda\) (the Lagrangian) an \((n+1)\)-form \(E_\lambda\) (the Euler-Lagrange form) by using the exterior differential operator \(d\), an appropriate canonical decomposition of the underlying spaces of forms and the concept of a Lepage form, one can include the Euler-Lagrange mapping in a differential sequence of sheaves. It follows the possibility to study the global properties of the Euler-Lagrange mapping. After explaining some elements of the sheaf theory, insisting on those which are important for the variational structures, the author studies the critical points of the variational functionals and some geometric problems connected to the various kinds of symmetries of the variational functionals. Several variational principles in physical field theory and geometric mechanics are obtained as particular cases of the general theory. The last chapter (8) is devoted to the study of the variational sequence of order \(r\) for a fibered manifold \(Y\). Its construction is based on the fact that the de Rham sequence of differential forms on the \(r\)-jet prolongation \(J^rY\) has a remarkable subsequence defined by the contact forms. The variational sequence is obtained as the quotient sheaf of de Rham sequence. The author considers some local properties of the terms of the quotient sequence and global properties represented by theorems on the cohomology of the complex of global sections of the variational sequence.

Reviewer: Vasile Oproiu (Iaşi)

##### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

49Q20 | Variational problems in a geometric measure-theoretic setting |

58E20 | Harmonic maps, etc. |

58E30 | Variational principles in infinite-dimensional spaces |