Introduction
This book is an academic monograph on the calculus of variations for teaching, the
fundamental theories and their applications of the calculus of variations are
systematically introduced in it.
The purpose of writing the book is that the authors hope to provide a textbook or
teaching reference book of the variational method course for the graduate students
and higher grade students in colleges and universities, enable them to be familiar
with the basic concepts and the calculation methods of the calculus of variations.
The contents in this book include the preliminaries, variational problems with fixed
boundaries, sufficient conditions of extrema of functionals, problems with undetermined 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. Many of the
contents are the research achievements of the authors for many years, especially the
extremum function theorem of complete functional is proposed, all kinds of Euler
equations in the variational methods are unified, the connotation of the Hilbert
adjoint operator is extended, the definitions of the other three kinds of adjoint
operators are proposed, the variational theories of functional with vector, modulus
of vector, arbitrary order tensor, trace of tensor, transposed tensor, Hamiltonian
operators and Hamiltonian operator string are founded, corresponding the Euler
equations and the natural boundary conditions are presented, the application range
of the variational methods is expanded. This book can also be used as reference for
the relevant professional teachers, science and technology personnel.
In this book, the concepts are clear, the logic is rigorous and the contents are
rich, the profound questions are explained in plain terms, it is easy to self-study.
Pay attention to the introduction of methods, and do not break the abstractness,
systematicness, scientificity and rigorousness of mathematics. The book has a large
number of examples and exercises, and with index. In order to help the readers to
solve the difficulties encountered in the study, the book gives the chapters all
solving process of more than 300 problems and answers for the reader’s reference.
xi
This book is an academic monograph on the calculus of variations for teaching, the
fundamental theories and their applications of the calculus of variations are
systematically introduced in it.
The purpose of writing the book is that the authors hope to provide a textbook or
teaching reference book of the variational method course for the graduate students
and higher grade students in colleges and universities, enable them to be familiar
with the basic concepts and the calculation methods of the calculus of variations.
The contents in this book include the preliminaries, variational problems with fixed
boundaries, sufficient conditions of extrema of functionals, problems with undetermined 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. Many of the
contents are the research achievements of the authors for many years, especially the
extremum function theorem of complete functional is proposed, all kinds of Euler
equations in the variational methods are unified, the connotation of the Hilbert
adjoint operator is extended, the definitions of the other three kinds of adjoint
operators are proposed, the variational theories of functional with vector, modulus
of vector, arbitrary order tensor, trace of tensor, transposed tensor, Hamiltonian
operators and Hamiltonian operator string are founded, corresponding the Euler
equations and the natural boundary conditions are presented, the application range
of the variational methods is expanded. This book can also be used as reference for
the relevant professional teachers, science and technology personnel.
In this book, the concepts are clear, the logic is rigorous and the contents are
rich, the profound questions are explained in plain terms, it is easy to self-study.
Pay attention to the introduction of methods, and do not break the abstractness,
systematicness, scientificity and rigorousness of mathematics. The book has a large
number of examples and exercises, and with index. In order to help the readers to
solve the difficulties encountered in the study, the book gives the chapters all
solving process of more than 300 problems and answers for the reader’s reference.
xi