zbMATH — the first resource for mathematics

Generalized classical mechanics and field theory. A geometrical approach of Lagrangian and Hamiltonian formalisms involving higher order derivatives. (English) Zbl 0581.58015
North-Holland Mathematics Studies, 112. Notas de matemática (102). Amsterdam - New York - Oxford: North-Holland. XV, 289 p. $ 44.50; Dfl. 120.00 (1985).
The term ”generalized” that appears in the title of this book would seem to refer to the modern development of particle mechanics and classical field theory in terms of differential geometric methods on manifolds on the one hand, and the inclusion of variational principles of higher order on the other, that is, the admission of Lagrangians that may depend on derivatives of order greater than the first of the field variables. In the opinion of the authors an appropriate arena for this analysis is provided by the theory of jet bundles, and accordingly the first chapter is devoted to an introduction to jet manifolds. As a particular case of the latter the tangent bundle of higher order is considered, with special emphasis on the so-called almost tangent structure of higher order, which is regarded as a generalization of the almost tangent geometry of the tangent bundle. In a similar vein the notion of a spray of higher order is introduced. These concepts are applied in the second chapter to a formalism designed to cope with higher order Lagrangians that depend on a single independent variable in a manner that makes use of some techniques of symplectic geometry (a brief description of the latter having been given). This leads to a discussion of the associated hamiltonian formalism, including Poisson brackets, canonical transformations, and the Hamilton-Jacobi equation.
The third and final chapter is devoted to field theory; it begins with some comments concerning the well-known difficulties that are inherent in the formulation of a suitable canonical formalism for multiple integral variational problems. A description of the standard field-theoretic approach culminates in the de Donder-Hamilton equations. This is followed by a detailed analysis of a corresponding formalism on jet manifolds and various important ramifications thereof. The book concludes with two appendices, of which the first deals with vector bundles, while the second is concerned with constraints and presymplectic systems.
According to the foreword ”this book is addressed mainly to graduate students”. Because of a presentation that would appear to be somewhat uneven as regards mathematical sophistication, the reviewer is not entirely convinced that this is a totally realistic objective. Moreover, the reader is occasionally confronted with disconcerting statements such as the following on p. 243: ”As we know, in order to obtain a Hamiltonian formalism in Analytical Mechanics we need all momentum variables to be independent of the velocity variables.” Although the bibliography is fairly extensive, it is somewhat random and contains various inaccuracies, including at least one incorrect attribution of authorship.
Reviewer: H.Rund

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
70-02 Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems
58A20 Jets in global analysis
53C80 Applications of global differential geometry to the sciences
58E30 Variational principles in infinite-dimensional spaces