Calculus of Variations

Description

29. The Maxwell equations are actually invariant under a 15-parameter family (group) of transformations. In addition to the 10 conservation laws already mentioned (energy, momentum and angular momentum), this invariance leads to 5 more conservation laws, which, however, do not have direct physical meaning. For a detailed treatment of this problem, see E. Bessel-Hagen, Über die Erhaltungssätze der Elektrodynamik , Math. Ann., 84 , 258 (1921).

cf. Noether 2010 for the history of the reception of Noether's theorem, which cites this book on p. 105 (PDF p. 118).

Based on a series of lectures given by I. M. Gelfand at Moscow State University, this book actually goes considerably beyond the material presented in the lectures. The aim is to give a treatment of the elements of the calculus of variations in a form both easily understandable and sufficiently modern. Considerable attention is devoted to physical applications of variational methods, e.g., canonical equations, variational principles of mechanics, and conservation laws. The reader who merely wishes to become familiar with the most basic concepts and methods of the calculus of variations need only study the first chapter. Students wishing a more extensive treatment, however, will find the first six chapters comprise a complete university-level course in the subject, including the theory of fields and sufficient conditions for weak and strong extrema. Chapter 7 considers the application of variational methods to the study of systems with infinite degrees of freedom, and Chapter 8 deals with direct methods in the calculus of variations. The problems following each chapter were made specially for this English-language edition, and many of them comment further on corresponding parts of the text. Two appendices and suggestions for supplementary reading round out the text. Substantially revised and corrected by the translator, this inexpensive new edition will be welcomed by advanced undergraduate and graduate students of mathematics and physics.