Foreword
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. Lao’s 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. 2 presents the variational problem of the complete functional,
and the relevant theorems and their proofs. Chapter 7 gives the other three kinds of
definitions about the adjoint operator, and the connotation of Hilbert adjoint
operator, and Chap. 10 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
v
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. Lao’s 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. 2 presents the variational problem of the complete functional,
and the relevant theorems and their proofs. Chapter 7 gives the other three kinds of
definitions about the adjoint operator, and the connotation of Hilbert adjoint
operator, and Chap. 10 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
v