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

In order to discuss the convenience of problems, here are some concepts related to the variational methods.

A collection of functions that have some common properties is called the class function, function class or class of functions, it is written as $F$. For instance, in Example 2.1.1, All of the plane curves pass through point $A$ and point $B$, therefore every function passes through point $A$ and point $B$ is the common property of the function set. In Example 2.1.3, the perimeters of all smooth closed curves are given $L, L$ is a common property of the function set.
The common class functions have:
The continuous function set in the open interval $\left(x_0, x_1\right)$, it is called the continuous function class in an open interval $\left(x_0, x_1\right)$, it is written as $C\left(x_0, x_1\right)$.

The continuous function set in the closed interval $\left[x_0, x_1\right]$, it is called the continuous function class in a closed interval $\left[x_0, x_1\right]$, it is written as $C\left[x_0, x_1\right]$, where the function at the left endpoint of the interval is right continuous, at the right endpoint of the interval is left continuous. At this moment the continuity at an endpoint of the interval is called one-sided continuity.

In an open interval $\left(x_0, x_1\right)$, the function set that $n$th derivative is continuous is called the function class of continuous $n$th derivative in an open interval $\left(x_0, x_1\right)$, it is written as $F=\left{y(x) \mid y \in C^n\left(x_0, x_1\right)\right.$ or $C^n\left(x_0, x_1\right)$, and stipulate $C^0\left(x_0, x_1\right)=C\left(x_0, x_1\right)$, namely the zero-order derivative of the function class is the function class itself.

The symbol $C^n\left(x_0, x_1\right)$ denotes that the function set in a neighborhood of each point in the open interval $\left(x_0, x_1\right)$ is defined, in $\left(x_0, x_1\right)$ there is the set of all the functions $y(x)$ of $\leq n$ all-order continuous derivatives. If for each $n$, there is $y(x) \in$ $C^n\left(x_0, x_1\right)$, then $y(x)$ is called the infinite differentiable function, it is written as $y(x) \in C^n\left(x_0, x_1\right) \equiv \bigcap_{n=0}^{\infty} C^n\left(x_0, x_1\right)$.

In a closed interval $\left[x_0, x_1\right]$, the function set that $n$th derivative is continuous is called the function class of continuous $n$th derivative in a closed interval $\left[x_0, x_1\right]$, it is written as $F=\left{y(x) \mid y \in C^n\left[x_0, x_1\right], y\left(x_0\right)=y_0, y\left(x_1\right)=y_1\right}$ or $C^n\left[x_0, x_1\right]$, where the $n$th derivatives of the function $y$ are one-sided continuity at the endpoints of the interval, $y_0$ and $y_1$ are fixed constants, and stipulate $C^0\left[x_0, x_1\right]=C\left[x_0, x_1\right]$.
In an open interval $\left(x_0, x_1\right)$ and closed interval $\left[x_0, x_1\right]$, the function set that $n$th derivative is continuous can all be written as $F={y(x)}$.

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

Let $F\left(x, y(x), y^{\prime}(x)\right)$ be a given function of these three independent variables $x$, $y(x), y^{\prime}(x)$ in the interval $\left[x_0, x_1\right]$, and second-order continuously differentiable, where, $y(x)$ and $y^{\prime}(x)$ are unknown functions of $x$, then the functional
$$J[y(x)]=\int_{x_0}^{x_1} F\left(x, y(x), y^{\prime}(x)\right) \mathrm{d} x$$
is called the simplest functional of integral type or simplest integral type functional, it is called the simplest functional for short, sometimes it is also called the cost functional. The functional $J[y(x)]$ is called the functional form or variational integral. The integrand $F$ is called the kernel of a functional, variational integrand, variational integrand function, Lagrange function or Lagrangian function. Because the value of $J[y(x)]$ obtained integral with respect to $F$ depends on the form of the function $y(x)$, therefore $J[y(x)]$ is the functional of $y(x)$. The functional (2.3.1) shows that $J[y(x)]$ is actually not only the function of $y(x)$, but also the function of $x$ and $y^{\prime}(x)$, but as long as $y(x)$ is found out, $y^{\prime}(x)$ can also be found out, thus the functional (2.3.1) only is written in the form of $J[y(x)]$.

In the one-order neighborhood of $y=y(x)$, taking an arbitrary curve $y=y_1(x)$, then there is
$$\delta y=y_1(x)-y(x), \delta y^{\prime}=y_1^{\prime}(x)-y^{\prime}(x)$$
The increment of the simplest functional $J[y]=\int_{x_0}^{x_1} F\left(x, y, y^{\prime}\right) \mathrm{d} x$ is
\begin{aligned} \Delta J &=J\left[y_1\right]-J[y]=J[y+\delta y]-J[y] \ &=\int_{x_0}^{x_1} F\left(x, y+\delta y, y^{\prime}+\delta y^{\prime}\right) \mathrm{d} x-\int_{x_0}^{x_1} F\left(x, y, y^{\prime}\right) \mathrm{d} x \ &=\int_{x_0}^{x_1}\left[F\left(x, y+\delta y, y^{\prime}+\delta y^{\prime}\right)-F\left(x, y, y^{\prime}\right)\right] \mathrm{d} x \end{aligned}
Sometimes the increment of a functional is also called the total variation of the functional.

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

Ĵ[是(X)]=∫X0X1F(X,是(X),是′(X))dX

d是=是1(X)−是(X),d是′=是1′(X)−是′(X)

DĴ=Ĵ[是1]−Ĵ[是]=Ĵ[是+d是]−Ĵ[是] =∫X0X1F(X,是+d是,是′+d是′)dX−∫X0X1F(X,是,是′)dX =∫X0X1[F(X,是+d是,是′+d是′)−F(X,是,是′)]dX

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