数学代写|变分法代写Calculus of Variations代考|MATH655

数学代写|变分法代写Calculus of Variations代考|The Euler Equations of the Simplest Functional

Theorem 2.4.1 Let the simplest functional
$$J[y(x)]=\int_{x_0}^{x_1} F\left(x, y, y^{\prime}\right) \mathrm{d} x$$
obtain extremum and satisfy the fixed boundary conditions
$$y\left(x_0\right)=y_0, y\left(x_1\right)=y_1$$
then the extremal curve $y=y(x)$ should satisfy the following necessary condition
$$F_y-\frac{\mathrm{d}}{\mathrm{d} x} F_{y^{\prime}}=0$$
there, $F$ is the known function of $x, y, y^{\prime}$ and has the second continuous partial derivative.

Note that the second term on the left side of Eq. (2.4.3) is the total derivative to the independent variable $x$.

Equation (2.4.3) is called the Euler equation of the function (2.4.1), it was obtained by Swiss mathematician Euler in 1736 (otherwise in 1741 or in 1744). However, at that time Euler’s proof was some complicated, he deduced the Euler equation with the method of broken line approximating curve. Later the French mathematician Lagrange improved Euler’s proof in very concise way, and on August 12 , 1755 he told the proof in the form of a letter to Euler. Therefore Eq. (2.4.3) is also called the Euler-Lagrange(‘s) equation. Thus, the calculus of variations formed a new branch of mathematical analysis. The Euler equation is the variational condition of a functional in the domain. In the calculus of variations, all of the differential equations corresponding to the functional with the structure of Eq. (2.4.1) can be called the Euler equation.
The Euler equation can also be written as
$$F_y-F_{x y^{\prime}}-F_{y y^{\prime}} y^{\prime}-F_{y^{\prime} y^{\prime}} y^{\prime \prime}=0$$

数学代写|变分法代写Calculus of Variations代考|Several Special Cases of the Euler Equation

Because the various partial derivatives $F_y, F_{y^{\prime} y^{\prime}}, F_{y^{\prime} y}$ and $F_{y^{\prime} x}$ of $F$ in the Euler equation (2.4.4) may contain $x, y$ and $y^{\prime}$, in general, it is not a linear differential equation, so the equation often can not be simply solved, but when $F$ doesn’t explicitly contain one or two among $x, y$ and $y^{\prime}$, the problems are likely to get simplified. This section will discuss some special forms of the integrand $F\left(x, y, y^{\prime}\right)$ in the functional (2.4.1)
(1) $F$ does not depend on $y^{\prime}$ or only relies on $y$, namely $F=F(x, y)$ or $F=F(y)$
At this time, $F_{y^{\prime}} \equiv 0$, so the Euler equation $F_y(x, y)=0$ or $F_y(y)=0$, this is a function equation, the solution does not contain arbitrary constant. The solution of the function equation doesn’t satisfy the boundary conditions: $y\left(x_0\right)=y_0, y\left(x_1\right)=y_1$, the variational problem has no solution. In rare cases, for instance, only when the solution of $F_y(x, y)=0$ or $F_y(y)=0$ passes through points $\left(x_0, y_0\right)$ and $\left(x_1, y_1\right)$, it can become an extremal curve. If the problem has a solution, there will be no additional boundary conditions.

Example 2.5.1 Known the functional $J[y]=\pi \int_{x_0}^{x_1} y^2 \mathrm{~d} x, y\left(x_0\right)=y_0, y\left(x_1\right)=y_1$, find the extremum of the functional $J[y]$.

Solution The Euler equation of the functional is $2 y=0$ or $y=0$, if and only if $y_0=y_1=0, y=0$ the value of the functional $J[y]$ is minimum. Otherwise the minimum of the functional $J[y(x)]$ can not be reached in the continuous function class.

Example 2.5.2 Known the functional $J[y]=\int_0^\pi y(2 x-y) \mathrm{d} x, y(0)=0, y(\pi)=$ 1 , find the extremal curve of the functional.

数学代写|变分法代写Calculus of Variations代考|The Euler Equations of the Simplest Functional

$$J[y(x)]=\int_{x_0}^{x_1} F\left(x, y, y^{\prime}\right) \mathrm{d} x$$

$$y\left(x_0\right)=y_0, y\left(x_1\right)=y_1$$

$$F_y-\frac{\mathrm{d}}{\mathrm{d} x} F_{y^{\prime}}=0$$

$$F_y-F_{x y^{\prime}}-F_{y y} y^{\prime}-F_{y^{\prime} y^{\prime}} y^{\prime \prime}=0$$

数学代写|变分法代写Calculus of Variations代考|Several Special Cases of the Euler Equation

(1) $F$ 不依赖于 $y^{\prime}$ 或仅依靠 $y$ ，即 $F=F(x, y)$ 或者 $F=F(y)$

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