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# Fundamental Theories and Their Applications of the Calculus of Variations

This book focuses on the calculus of variations, including fundamental theories and applications.

This textbook is intended for graduate and higher-level college and university students, introducing them to the basic concepts and calculation methods used in the calculus of variations. Les mer
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This book focuses on the calculus of variations, including fundamental theories and applications.

This textbook is intended for graduate and higher-level college and university students, introducing them to the basic concepts and calculation methods used in the calculus of variations. It covers 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 for variational problems, variational principles in mechanics and their applications, and variational problems of functionals with vector, tensor and Hamiltonian operators. Many of the contributions are based on the authors' research, addressing topics such as the extension of the connotation of the Hilbert adjoint operator, definitions of the other three kinds of adjoint operators, the extremum function theorem of the complete functional, unified Euler equations in variational methods, variational theories of functionals with vectors, modulus of vectors, arbitrary order tensors, Hamiltonian operators and Hamiltonian operator strings, reconciling the Euler equations and the natural boundary conditions, and the application range of variational methods. The book is also a valuable reference resource for teachers as well as science and technology professionals.

Preliminaries.- 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.- Methods of Variational Problems.- Variational Principles in Mechanics and Their Applications.- Variational Problems of Functionals with Vector, Tensor and Hamiltonian Operators.

Dazhong Lao is an Associate Professor at the School of Aerospace Engineering, Beijing Institute of Technology, and a Distinguished Professor of Yinchuan University of Energy.

As an Associate Professor from 1998 to 2017, his research focused on combustion flow theory and testing technologies for rocket motors. During this time, he published more than 100 academic papers and presided over or participated in over 20 scientific research projects. Since August 2017, he has served as a Senior Engineer involved in thermal system modeling at Beijing Dafeng Tianli Technology Co., Ltd. He was appointed as a Distinguished Professor and Academic Leader of Energy and Power Engineering at Yinchuan University of Energy in 2018.

His interests include rubber product development; chemical engineering machinery; fluid machinery; energy and power engineering; combustion flow theory; design, testing and simulation technologies for rocket motors; theory, design and experimental technologies for propulsion systems; and applied mathematics.

Shanshan Zhao is a Senior Engineer at Dandong Chemical Fiber Company, Liaoning Province. She received her Bachelor's degree in Chemical Machinery from the Dalian Institute of Technology in 1982 and subsequently worked in the Equipment Installation Team at Dandong Chemical Fiber Company.

Her main research interests include chemical fiber machinery, pressure vessels, fluid machinery and applied mathematics. She is the co-author of one book and has published a number of papers on the calculus of variations.

She served as Director of the Equipment Department at the Design Institute of Dandong Chemical Fiber Company from 1994 to 2013. She was an assessor for the first and second-class pressure vessel design and appraisal for the Liaoning Province Quality and Technical Supervision Association from 2002 to 2015, and for the Special Equipment Safety Technology Association from 2015 to 2018.