Dazhong Lao - Fundamental Theories and Their Applications of the Calculus of Variations

Jointly published with Beijing Institute of Technology Press

The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Beijing Institute of Technology Press.

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The calculus of variations is an important branch of applied mathematics, which deals with the extremum problem of a functional in the integral form. This branch has found wide applications to mathematical and physical sciences, as well as a great variety of engineering fields.

Mr. Dazhong Lao, the first author of this book, is an Associate Professor with School of Aerospace Engineering, Beijing Institute of Technology, while Ms. Shanshan Zhao, the co-author of the book, is a Senior Engineer. Professor Lao has been engaged in studying the calculus of variations over the past three decades. His experience of teaching the calculus of variations has also exceeded 20 years. As early as 2004, he had his book entitled Fundamentals of the Calculus of Variations published by The Defense Industry Publishing House in China. Later on, the second edition and the third edition of the book were published in 2007 and 2015, respectively. His book is so excellent that the total print number of the above three editions is over ten thousand. Professor Lao gave me the third edition of the book as soon as it became available in January 2015. I greatly enjoyed reading the excellent book. To my best knowledge, this is the most comprehensive book regarding to the topic of calculus of variations with numerous examples included.

The current book to be published by Springer Verlag is a further extension of Prof. Laos previous book in the frame of theories and their Applications of the Calculus of Variations. The book presents many original ideas of the basic theories of the calculus of variations, including some research achievements of the authors. For example, Chap. gives some examples of the four kinds of adjoint operators. In that chapter, the authors made joint efforts to put forward the fundamental lemma of the variation of functional with tensors, to find the variational theory of functional with vector, modulus of vector, tensor, trace of tensor, transposed tensor, Hamiltonian operator and Hamiltonian operator string. They also gave the Euler equations and the corresponding natural boundary conditions, and a large number of examples. The above theories will bring great convenience to readers so as to gain an insight into the calculus of variations and to apply the methods to complicated problems of concern.

Hence, the current book is not only an ideal textbook for graduate students and undergraduate students of senior grade, but also a monograph for researchers in various fields. I am sure that the book will expose the achievements of my two colleagues to the international community of applied mathematics and promote the academic exchanges.

The calculus of variations is a branch of mathematics developed at the end of the seventeenth century. In a sense, it can be said that the calculus of variation is a science refined from the other sciences. This book mainly introduces the classical calculus of variations, its theory is complete, it has a wide range of applications in many aspects, such as mechanics, physics, chemistry, optics, acoustics, heat transfer, materials science, tribology, economics, aerospace theory, information theory, automatic control theory, image processing, medicine and biology, etc. The finite element method developed in the middle of the twentieth century, one of its mathematical foundations is the calculus of variations. Now the calculus of variations has become an essential mathematical foundation of graduate students, college students, engineers, technicians and scientific workers.

The purpose of writing this book is to provide a textbook or teaching reference book learning the calculus of variations for the graduate students and senior grade college students of institutions of higher learning, such that them can be familiar with the basic concepts and calculation methods of the calculus of variations, in which it includes the preliminary knowledge, variational problems with fixed boundaries, sufficient conditions of extrema of functionals, problems with variable boundaries, variational problems of conditional extrema, variational problems in parametric forms, variational principles, direct methods of variational problems, variational principles in mechanics and their applications and variational problems of functionals with vector, tensor and Hamiltonian operators. Quite a part of contents in this book are the authors research achievements. Especially the authors proposed the extremal function theorem of the complete functional, unified various kinds of Euler equations in the calculus of variations, originated the variational theory of functionals with vector, modulus of vector, arbitrary order tensor and Hamiltonian operators, gave the corresponding Euler equations and natural boundary conditions, expanded the application range of the calculus of variations. This book can also be used as relevant professional teachers, scientific research personnel and engineering technical personnels reference.