I enjoyed reading The calculus of variations. As such, it is intended to be a non-intimidating, introductory text …. "The calculus of variations is one of the latest books in Springer’s Universitext series. Stachó, Acta Scientiarum Mathematicarum, Vol. It can also be appreciated that the author tries to present the results showing motivation and heuristical ideas for each crucial theorem." (L. According to my classroom experience with undergraduate physicists, the presentation of the examples in the book may be very helpful …. It is written with a deep pedagogical attention …. "I find this book a very useful supplementary reading for undergraduate students and a good teaching aid for lecturers of topics involving traditional variational calculus (as e. The student can thus learn the main results in each chapter and return as needed to the proofs for a deeper understanding. The technical details for many of the results can be skipped on the initial reading. More importantly, the book is written on two levels. In addition, topics such as Hamilton’s Principle, eigenvalue approximations, conservation laws, and nonholonomic constraints in mechanics are discussed. For the reader interested mainly in techniques and applications of the calculus of variations, I leavened the book with numerous examples mostly from physics. I have made “passive use” of functional analysis (in particular normed vector spaces) to place certain results in context and reassure the mathematician that a suitable framework is available for a more rigorous study. I have paused at times to develop the proofs of some of these results, and discuss briefly various topics not normally found in an introductory book on this subject such as the existence and uniqueness of solutions to boundary-value problems, the inverse problem, and Morse theory. The reader interested primarily in mathematics will find results of interest in geometry and differential equations. This book is an introduction to the calculus of variations for mathematicians and scientists. Much of the mathematics underlying control theory, for instance, can be regarded as part of the calculus of variations. More recently, the calculus of variations has found applications in other fields such as economics and electrical engineering. Substantially revised and corrected by the translator, this inexpensive new edition will be welcomed by advanced undergraduate and graduate students of mathematics and physics.The calculus of variations has a long history of interaction with other branches of mathematics such as geometry and differential equations, and with physics, particularly mechanics. Two appendices and suggestions for supplementary reading round out the text. 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. 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. 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. 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. Considerable attention is devoted to physical applications of variational methods, e.g., canonical equations, variational principles of mechanics, and conservation laws. The aim is to give a treatment of the elements of the calculus of variations in a form both easily understandable and sufficiently modern. Gelfand at Moscow State University, this book actually goes considerably beyond the material presented in the lectures. Based on a series of lectures given by I.
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