Chapter 2
Variational Problems with Fixed
Boundaries
The calculus of variations is also called the variational methods or variational
calculus, it is a branch of mathematical analysis began to grow at the end of the
17th century, it is a science to study the definite integral type extremum of a functional which depends on some unknown functions. In short, the method to find the
extremum of a functional is called the calculus of variations. The problem to find
the extremum of a functional is called the variational problem, variation problem
or variational principle. On February 5, 1733, Clairaut published the first treatise
of the variational methods Sur quelques questions de maximis et minimis. The work
published by Euler in 1744 Methodus Inveniendi Lineas Curvas Maximi Minimive
Proprietate Gaudentes, Sive Solution Problematis Isoperimetrici Latissimo sensu
Accepti (A method of finding curves which have a property of extremes or solution
of the isoperimetric problem, if it is understood in the broadest sense of the word)
marked the birth of the calculus of variations as a new branch of mathematics. The
word variational methods was proposed for the first time by Lagrange in August
1755 in a letter to Euler, He called it the method of variation, while Euler in a paper
in 1756 proposed the word the calculus of variation. The name of the variational
method subject was obtained thereout.
The calculus of variations is an important part of functional analysis, but calculus
of variations appeared first, and the functional analysis appeared later.
This chapter through the examples of several classical variational problems illustrates the concepts of functional and variation. Mainly discuss the first variation
of functional, necessary conditions of extremum of a functional, Euler equations,
special cases of Euler equations, and solving methods of different types of extremal
functions of functionals under the fixed boundary conditions, particularly discuss the
variational problems of the complete functional.
© Beijing Institute of Technology Press and Springer Nature Singapore Pte Ltd. 2021
D. Lao and S. Zhao, Fundamental Theories and Their Applications of the Calculus
of Variations, https://doi.org/10.1007/978-981-15-6070-5_2
85
Variational Problems with Fixed
Boundaries
The calculus of variations is also called the variational methods or variational
calculus, it is a branch of mathematical analysis began to grow at the end of the
17th century, it is a science to study the definite integral type extremum of a functional which depends on some unknown functions. In short, the method to find the
extremum of a functional is called the calculus of variations. The problem to find
the extremum of a functional is called the variational problem, variation problem
or variational principle. On February 5, 1733, Clairaut published the first treatise
of the variational methods Sur quelques questions de maximis et minimis. The work
published by Euler in 1744 Methodus Inveniendi Lineas Curvas Maximi Minimive
Proprietate Gaudentes, Sive Solution Problematis Isoperimetrici Latissimo sensu
Accepti (A method of finding curves which have a property of extremes or solution
of the isoperimetric problem, if it is understood in the broadest sense of the word)
marked the birth of the calculus of variations as a new branch of mathematics. The
word variational methods was proposed for the first time by Lagrange in August
1755 in a letter to Euler, He called it the method of variation, while Euler in a paper
in 1756 proposed the word the calculus of variation. The name of the variational
method subject was obtained thereout.
The calculus of variations is an important part of functional analysis, but calculus
of variations appeared first, and the functional analysis appeared later.
This chapter through the examples of several classical variational problems illustrates the concepts of functional and variation. Mainly discuss the first variation
of functional, necessary conditions of extremum of a functional, Euler equations,
special cases of Euler equations, and solving methods of different types of extremal
functions of functionals under the fixed boundary conditions, particularly discuss the
variational problems of the complete functional.
© Beijing Institute of Technology Press and Springer Nature Singapore Pte Ltd. 2021
D. Lao and S. Zhao, Fundamental Theories and Their Applications of the Calculus
of Variations, https://doi.org/10.1007/978-981-15-6070-5_2
85