## 数学代写|变分法代写Calculus of Variations代考|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

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.

## 数学代写|变分法代写Calculus of Variations代考|Examples of the Classical Variational Problems

The basic problem of variational methods is to find the extremal problems of functionals and the corresponding extremal functions. In order to show the research contents of variational methods, first embark from the several classical variational examples to raise the concept of functional.

Example 2.1.1 The brachistochrone problem, problem of brachistochrone or problem of curve of steepest descent. This is one of the earliest appeared variational problems in the history, which was usually considered the beginning of the history of the variational methods, was also a symbol of the development of the variational methods. It was first proposed by Galileo in 1630, he systematically studied the problem again in 1638 , but at that time he gave the wrong results, he thought this was a circular arc curve. The substantial research of variational method was that John Bernoulli wrote to his brother Jacob Bernoulli an open letter on the Leipziger Acta Eurditorum in the June 1696 issue to ask for the solution to the problem. The formulation of the problem was: Assuming that $A$ and $B$ are the two points which are not in the same vertical straight line in a vertical plane, in all the plane curves joining point $A$ and point $B$, determine a curve, such that the time needed is the shortest when a particle that is acted on only by gravity and the initial velocity is zero moves from point $A$ to point $B$ along the curve. This problem had caused many mathematicians’ interest at that time. After Newton heard the news on January 29, 1697, he solved this problem on the same day. Leibniz, Bernoulli brothers and L’Hospital et al. all studied this problem, they obtained correct results in different ways, among them, Jacob Bernoulli started from geometric intuition, he gave the more general solution, the solution took a big step towards the direction of the variational methods. Except Jacob Bernoulli’s method of solution, others’ methods of solution were published on the Acta eurditorum in the May 1697 issue.

Solution The particle motion time depends not only on the length of the path, but also is associated with the speed. In all the curves joining point $A$ and point $B$, the straight line distance $A B$ is the shortest (see the solution of Example 2.5.11), but it is not necessarily a particle motion time shortest path. Now to establish the mathematical model of this problem. As shown in Fig. 2.1, taking $A$ as the origin of plane rectangular coordinate system, $x$ axis is put in a horizontal position, the direction of $y$ axis is downward. Obviously, the brachistochrone should be in the plane. Thus the coordinate of point $A$ is $(0,0)$. Let the coordinate of point $B$ be $\left(x_1, y_1\right)$, the equation of a curve joining point $A$ and point $B$ is
$$y=y(x) \quad\left(0 \leq x \leq x_1\right)$$

## 数学代写|变分法代写Calculus of Variations代考|Examples of the Classical Variational Problems

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