The concept and operation rules of Calculus of variations (2)

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6. Euler-Lagrange equation

6.6 Generalization

• Univariate Single Functions with Higher Derivatives

The stationary value of the functional:

$$I[f]={\int }_{x_0}^{x_1}\mathcal{L}(x,f,f^{\prime },f^{\prime \prime },\dots ,f^{(k)})\mathrm{d}x;f^{\prime }:=\frac{\mathrm{d}f}{\mathrm{d}x},f^{\prime \prime }:=\frac{{\mathrm{d}}^2}{d{x}^2},f^{(k)}:=\frac{d^{k}f}{d{x}^k}$$

It can be obtained from the Euler-Lagrangian equation:

$$\frac{\partial \mathcal{L}}{\partial f}-\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\partial L}{\partial f^{\prime }}\right)+\frac{{\mathrm{d}}^2}{d{x}^2}\left(\frac{\partial L}{\partial f^{\prime \prime }}\right)-\cdots +(-1{)}^k\frac{d^k}{d{x}^k}\left(\frac{\partial L}{\partial f^{(k)}}\right)=0$$

In the function itself and the first $k-1$ derivative under fixed boundary conditions (i.e. for allf ( i ) , i ∈ { 0 , . . . , k − 1 } f^{(i)},i\in \{0,... ,k- 1\}f(i),i∈{ 0,...,k−1 } ). Highest derivative$f^{The\; endpoint\; values\; for\; (k)}$ remain flexible.

• Multiple Functions of Single Variable with Single Derivative

If the problem involves finding a single independent variable ( x ) (x)Multiple functions of$($$x$$)$$(f_1,f_2,\dots ,f_m)$ , which are all on the extrema of the functional:

$$I[f_1,f_2,\dots ,f_m]={\int }_{x_0}^{x_1}\mathcal{L}(x,f_1,f_2,\dots ,f_m,f_1^{\prime },f_2^{\prime },\dots ,f_m^{\prime })\mathrm{d}x;f_i^{\prime }:=\frac{\mathrm{d}{f}_i}{\mathrm{d}x}$$

那么对应的欧拉-拉格朗日方程为：

$$\begin{array}{r}\frac{\partial \mathcal{L}}{\partial f_i}-\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\partial L}{\partial f_i^{\prime }}\right)=0;i=1,2,...,m\end{array}$$

• a single function of many variables with a single derivative

Multidimensional generalization comes from considering $function\; of\; n$ variables. If$\Omega$ is some surface, then:

$$I[f]={\int }_{\Omega }\mathcal{L}(x_1,\dots ,x_n,f,f_1,\dots ,f_n)\mathrm{d}\mathbf{x};f_j:=\frac{\partial f}{\partial x_j}$$

仅当 $f$ 满足偏微分方程时才被取极值：

$$\frac{\partial \mathcal{L}}{\partial f}-\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(\frac{\partial L}{\partial f_j}\right)=0$$

when $n=2$ and the functional$When\; I$ is the energy functional, this leads to the minimal surface problem of the soap film (Minimal surface).

• Multiple Functions of Multiple Variables with Single Derivative

If you want to determine several multivariate unknown functions such that

$$I[f_1,f_2,\dots ,f_m]={\int }_{}\mathcal{L}(x_1,\dots ,x_n,f_1,\dots ,f_m,f_{1,1},\dots ,f_{1,n},\dots ,f_{m,1},\dots ,f_{m,n})\mathrm{d}\mathbf{x};f_{i,j}:=\frac{\partial f_i}{\partial x_j}$$

Then the Euler-Lagrange equation is:

$$\begin{array}{rl}\frac{\partial \mathcal{L}}{\partial f_1}-\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(\frac{\partial L}{\partial f_{1,j}}\right)& =0_1\\ \frac{\partial L}{\partial f_2}-\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(\frac{\partial L}{\partial f_{2,j}}\right)& =0_2\\ ⋮⋮& ⋮\\ \frac{\partial L}{\partial f_m}-\sum _{j=1}^n\frac{\partial }{\partial x_j}\left(\frac{\partial L}{\partial f_{m,j}}\right)& =0_m.\end{array}$$

• Single function of two variables with higher derivative

If you want to determine an unknown function $f$ , which depends on two variables$x_1$and $x_2$, and if the function depends on ffhigher derivatives of$f$$n$ , such that

$$\begin{array}{rl}I[f]& ={\int }_{}\mathcal{L}(x_1,x_2,f,f_1,f_2,f_{11},f_{12},f_{22},\dots ,f_{22\dots 2})\mathrm{d}\mathbf{x}\\ & f_i:=\frac{\partial f}{\partial x_i},f_{ij}:=\frac{{\partial }^2}{\partial x_i\partial x_j},\dots \end{array}$$

Then the Euler-Lagrange equation is:

$$\begin{array}{rl}\frac{\partial \mathcal{L}}{\partial f}& -\frac{\partial }{\partial x_1}\left(\frac{\partial L}{\partial f_1}\right)-\frac{\partial }{\partial x_2}\left(\frac{\partial L}{\partial f_2}\right)+\frac{{\partial }^2}{\partial x_1^2}\left(\frac{\partial L}{\partial f_{11}}\right)+\frac{{\partial }^2}{\partial x_1\partial x_2}\left(\frac{\partial L}{\partial f_{12}}\right)+\frac{{\partial }^2}{\partial x_2^2}\left(\frac{\partial L}{\partial f_{22}}\right)\\ & -\cdots +(-1{)}^n\frac{{\partial }^n}{\partial x_2^n}\left(\frac{\partial L}{\partial f_{22\dots 2}}\right)=0\end{array}$$

This can be expressed simply as:

$$\frac{\partial L}{\partial f}+\sum _{j=1}^n\sum _{m_1\le \dots \le {\mu }_j}(-1{)}^j\frac{{\partial }^j}{\partial x_{m_1}\dots \partial x_{m_j}}\left(\frac{\partial \mathcal{L}}{\partial f_{m_1\dots m_j}}\right)=0$$

where $m_1\dots m_j$are the indices of the number of variables, that is, they go from $1$ to$2$ . Here for$m_1\dots m_j$The summation of the indices is only for $m_1\le m_2\le \dots \le m_j$, this is to avoid calculating the same partial derivative multiple times, for example $f_{12}=f_{21}$appears only once in the previous equation.

• Multiple functions of multiple variables with higher derivatives

if there is $p$ depends on$m$ variables$x_1,\cdots ,x_m$The unknown function $f_i$is determined, and if the function depends on $f_i$Higher derivatives of up to the nnthorder$n\; such\; that:$

$$\begin{array}{rl}I[f_1,\dots ,f_p]& ={\int }_{}\mathcal{L}(x_1,\dots ,x_m;f_1,\dots ,f_p;f_{1,1},\dots ,f_{p,m};f_{1,11},\dots ,f_{p,mm};\dots ;f_{p,1\dots 1},\dots ,f_{p,m\dots m})\mathrm{d}\mathbf{x}\\ & f_{i,}:=\frac{\partial f_i}{\partial x_{}},f_{i,m_1{m}_2}:=\frac{{\partial }^2{f}_i}{\partial x_{m_1}\partial x_{m_2}},\dots \end{array}$$

where $m_1\dots m_j$is the index of the number of variables, i.e. they go from $1$ to$m$ . Then the Euler-Lagrange equation is:

$$\frac{\partial \mathcal{L}}{\partial f_i}+\sum _{j=1}^n\sum _{m_{1}leq\dots \le {\mu }_j}(-1{)}^j\frac{{\partial }^j}{\partial x_{m_1}\dots \partial x_{m{u}_j}}\left(\frac{\partial \mathcal{L}}{\partial f_{i,m_1\dots m_j}}\right)=0$$

where $m_1\dots m_j$The summation avoids computing the same derivative $f_{i,m_1{m}_2}=f_{i,m_2{m}_1}$Multiple times, just like the previous section. This can be expressed more compactly as:

$$\sum _{j=0}^n\sum _{m_1\le \dots \le {\mu }_j}(-1{)}^j{\partial }_{m_1\dots m_j}^j\left(\frac{\partial \mathcal{L}}{\partial f_{i,m_1\dots m_j}}\right)=0$$

6.7 Generalization to Manifolds

Order $M$ is a smooth manifold, let$C^{\infty }([a,b])$ represents a smooth function space$f:[a,b]\to M.$ _ Then, for the functional$S:C^{\infty }([a,b])\to R$ form

$$S[f]={\int }_a^b(L\circ \dot{f})(t)$$

where $L:TM\to R$ is a Lagrangian function (Lagrangian), expressing$\mathrm{d}{S}_f=0$ is equivalent to the statement that for all$t\in [a,b]$，$\dot{f}$Trivialization ( trivialization)of each coordinate system in the neighborhood of$($$t$$)$$(x^i,X^i)$, yields the following$\dim M$ equation:

$$\forall i:\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial X^i}{|}_{\dot{f}(t)}=\frac{\partial L}{\partial x^i}{|}_{\dot{f}\left(t\right)}$$

6.8 Concrete examples

To illustrate this process, consider finding the extremum function $y=The\; problem\; of\; f\left(x\right)$ , which is to connect two points$(x_1,y_1)$和$(x_2,y_2)$ is the shortest curve. The arc length of the curve is given by:

$$To\left[and\right]={\int }_{x_1}^{x_2}\sqrt{1+[y^{\prime }(x){]}^2}\mathrm{d}x$$

and

$$y^{\prime }(x)=\frac{\mathrm{d}y}{dx},y_1=f(x_1),y_2=f\left(x_2\right)$$

Note that assuming $y$ is$A\; function\; of\; x$ loses generality; ideally, both should be functions of some other parameter. This method is for illustration purposes only.

Now the Euler-Lagrange equation will be used to find the extremum function $f\left(x\right)$ , which makes the function$A\left[y\right]$ min:

$$\frac{\partial L}{\partial f}-\frac{\mathrm{d}}{\mathrm{d}x}\frac{\partial L}{\partial f^{\prime }}=0$$

and

$$L=\sqrt{1+[f^{\prime }(x){]}^2}$$

due to $f$ does not appear explicitly in$In\; L$ , so the first term in the Euler-Lagrangian equation for all$f\left(x\right)$ are gone, so:

$$\frac{d}{dx}\frac{\partial L}{\partial f^{\prime }}=0$$

Substitute into $L$ expressions and derivatives:

$$\frac{\mathrm{d}}{\mathrm{d}x}\frac{f^{\prime }(x)}{\sqrt{1+[f^{\prime }(x){]}^2}}=0$$

therefore

$$\frac{f^{\prime }\left(x\right)}{\sqrt{1+\left[f^{\prime }\right(x){]}^2}}=c$$

$c$ is the integral constant. Then square the above formula left and right:

$$\frac{\left[f^{\prime }\right(x){]}^2}{1+\left[f^{\prime }\right(x){]}^2}=c^2$$

in:

$$0\le c^2<1$$

Solving the previous equation, we get:

$$\left[f^{\prime }\right(x){]}^2=\frac{c^2}{1-c^2}$$

this means

$$f^{\prime }(x)=m$$

is a constant, so the shortest curve connecting two points $(x_1,y_1)$和$(x_2,y_2)$ is:

$$f(x)=mx+b\text{with}m=\frac{y_2-y_1}{x_2-x_1}\text{and}b=\frac{x_2{y}_1-x_1{y}_2}{x_2-x_1}$$

So we find the extremum function $f\left(x\right)$ , which minimizes the functional$A\left[y\right]$ makes$A\left[f\right]$ is the minimum value. The equation of a line is$y=f\left(x\right)$ . In other words, in other words, the shortest distance between two points is a straight line.

7. Beltrami’s identity

The Beltrami identity, named after Eugenio Beltrami, is a special case of the Euler-Lagrange equation in the variational method.

The Euler-Lagrange equation is used to extremalize an action functional of the form:

$$I[y]={\int }_a^{b}L[x,y(x),y^{\prime }(x)]dx$$

aa $a$ and$b$ is a constant:

$$y^{\prime }(x)=\frac{dy}{dx}$$

If $\frac{\partial L}{\partial x}=0$ , the Euler-Lagrange equation is simplified to Beltrami identity:

$$L-f^{\prime }\frac{\partial L}{\partial f^{\prime }}=C$$

where $C$ is a constant. LLon the left$L$ relative to$f^{\prime }(x)$。

The intuition behind this result is that if the variable $x$ is actually time, then the statement$\frac{\partial L}{\partial x}=0$ means that the Lagrangian is time-independent. According toNoether's theorem, there is an associated conserved quantity. In this case, that quantity is the Hamiltonian, the Legendre transform of the Lagrange quantity, which (usually) coincides with the energy of the system. This in turn corresponds to (minus) the constant in Beltrami's identity.

• derive

根据链式法则，$L$ 的导数为

$$\frac{\mathrm{d}L}{\mathrm{d}x}=\frac{\partial L}{\partial x}\frac{\mathrm{d}x}{\mathrm{d}x}+\frac{\partial L}{\partial f}\frac{\mathrm{d}f}{\mathrm{d}x}+\frac{\partial L}{\partial f^{\prime }}\frac{\mathrm{d}{f}^{\prime }}{\mathrm{d}x}$$

因为 $\frac{\partial L}{\partial x}=0$，上式可以简化为：

$$\frac{\mathrm{d}L}{\mathrm{d}x}=\frac{\partial L}{\partial f}{f}^{\prime }+\frac{\partial L}{\partial f^{\prime }}{f}^{\prime \prime }$$

Then by combining it with the Euler-Lagrange equation:

$$\frac{\partial L}{\partial f}=\frac{\mathrm{d}}{dx}\frac{\partial L}{\partial f^{\prime }}$$

We get the following expression:

$$\frac{dL}{dx}=f^{\prime }\frac{d}{dx}\frac{\partial L}{\partial f^{\prime }}+f^{\prime \prime }\frac{\partial L}{\partial f^{\prime }}$$

According to the product rule, the right-hand side is equivalent to:

$$\frac{dL}{dx}=\frac{d}{dx}\left(f^{\prime }\frac{\partial L}{\partial f^{\prime }}\right)$$

By removing the differential and setting aside both sides of the equation, we obtain the Beltrami identity:

$$L-f^{\prime }\frac{\partial L}{\partial f^{\prime }}=C$$

• Solution to the brachistochrone problem

An example of an application of Beltrami identity is the brachistochrone problem, which involves finding the curve y = y ( x ) y=y(x) that minimizes the integral$y=y(x)$：

$$I[y]={\int }_0^a\sqrt{\frac{1+y^{\prime 2}}{y}}\mathrm{d}x$$

Integrand:

$$L(y,y^{\prime })=\sqrt{\frac{1+y^{\prime 2}}{y}}$$

explicitly does not depend on the integral variable $x$ , so the Beltrami identity applies:

$$L-y^{\prime }\frac{\partial L}{\partial y^{\prime }}=C$$

Substitute into $L$ and simplify:

$$y(1+y^{\prime 2})=1/C^2(const.)$$

It can be solved with results in parametric equation form:

$$\begin{gathered} x=A(\varphi -\sin )\_ \\ y=A(1-\cos )\_ \end{gathered}$$

$A$ is half of the above constant,$\frac{1}{2C^2}$, and $\varphi$ is a variable. These are the parametric equations for the cycloid.

8. Euler–Poisson equation

If $S$ depends ony ( x ) y(x)The higher order derivative of$y$$($$x$$)\; ,\; that\; is,\; if$

$$S={\int }_a^{b}f(x,y(x),y^{\prime }(x),\dots ,y^{(n)}(x))\mathrm{d}x$$

那么$y$ must satisfy the Euler-Poisson equation:

$$\frac{\partial f}{\partial y}-\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\partial f}{\partial y^{\prime }}\right)+\cdots +(-1{)}^n\frac{{\mathrm{d}}^n}{d{x}^n}\left[\frac{\partial f}{\partial y^{(n)}}\right]=0$$

9. Du Bois-Reymond's theorem

The discussion so far has assumed that the extremal function has two continuous derivatives, although the integral $The\; existence\; of\; J$ requires only the first derivative of the trial functions. The condition that the first variation disappears at the extrema can be regarded as a weak form of the Euler-Lagrangian equation. The Du Bois-Reymond theorem asserts that this weak form implies a strong form. If$L$ has continuous first and second derivatives with respect to all its parameters, and if

$$\frac{{\partial }^2}{\partial f^{\prime 2}}\ne 0$$

then $f$ has two continuous derivatives and satisfies the Euler-Lagrange equation.

10. Lavrentiev phenomenon

Hilbert was the first to provide good conditions for the Euler-Lagrange equations to give a stationary solution. In convex regions and positive cubic differentiable Lagrangians, the solution consists of countable parts that either move along the boundary or satisfy the Euler-Lagrange equations in the interior.

However, Lavrentiev showed in 1926 that in some cases there is no optimal solution but one can be arbitrarily close to it by increasing the number of sections. The Lavrentiev phenomenon identifies differences in the infimum of minimization problems between different classes of admissible functions. Take for example the following question, posed by Manià in 1934:[18]

{\displaystyle L[x]=\int _{0} ^{1}(x {3}-t) ^{2}x' {6},}{\displaystyle L[x]=\int _ {0} ^{{ 1}(x} {3}-t) ^{2}x' {6},}
{\displaystyle {A}={x\in W^{1,1}(0,1):x(0)=0 ,\ x(1)=1}.}{\displaystyle {A}= {x\in W^{1,1}(0,1):x(0)=0,\x(1)=1} .}
Obviously, {\displaystyle x(t)=t^{\frac {1}{3}}}{\displaystyle x(t)=t^{\frac {1}{3}}} minimizes the pan function, but we find any function {\displaystyle x\in W^{1,\infty }}{\displaystyle x\in W^{1,\infty }} gives a value far from the infimum.

示例（一维）传统上表现为 {\displaystyle W^{1,1}}W{ {1,1}} 和 {\displaystyle W^{1,\infty },}{\displaystyle W^ {1,\infty },} 但 Ball 和 Mizel[19] 获得了第一个在 {\displaystyle W^{1,p}}W{1,p} 和 {\displaystyle W^{1 ,q}}{\displaystyle W^{1,q}} 对于 {\displaystyle 1\leq p<q<\infty .}{\displaystyle 1\leq p<q<\infty .} 有几个结果给出不发生该现象的标准——例如“标准增长”、不依赖于第二个变量的拉格朗日函数或满足 Cesari 条件 (D) 的近似序列——但结果通常是特定的，适用于一小类泛函。

与拉夫连季耶夫现象相关的是排斥性：任何表现拉夫连季耶夫现象的泛函都会表现出弱排斥性。 [20]

11. Functions of several variables（多元函数）

例如，如果 {\displaystyle \varphi (x,y)}{\displaystyle \varphi (x,y)} 表示在 {\displaystyle x,y} x,y 平面，则其势能与其表面积成正比：

{\displaystyle U[\varphi ]=\iint _{D}{\sqrt {1+\nabla \varphi \cdot \nabla \varphi }},dx,dy.}{\displaystyle U[\varphi ]= \iint _{D}{\sqrt {1+\nabla \varphi \cdot \nabla \varphi }},dx,dy.}
Plateau 的问题包括找到一个函数，该函数在假设 {\displaystyle D}D 边界上的规定值的同时使表面积最小化；这些解称为最小曲面。这个问题的欧拉-拉格朗日方程是非线性的：
{\displaystyle \varphi _{xx}(1+\varphi _{y}^{2})+\varphi _{yy}(1+\varphi _{x}^{2})-2\varphi _{ x}\varphi _{y}\varphi _{xy}=0.}{\displaystyle \varphi _{xx}(1+\varphi _{y}^{2})+\varphi _{yy}(1 +\varphi _{x}^{2})-2\varphi _{x}\varphi _{y}\varphi _{xy}=0.}
有关详细信息，请参见 Courant (1950)。

狄利克雷原理
通常只考虑膜的小位移就足够了，其与无位移的能量差近似为

{\displaystyle V[\varphi ]={\frac {1}{2}}\iint _{D}\nabla \varphi \cdot \nabla \varphi ,dx,dy.}{\displaystyle V[\varphi ]= {\frac {1}{2}}\iint {D}\nabla \varphi \cdot \nabla \varphi ,dx,dy.}
Experimental function for specified values ​​on all assumptions{\displaystyle D.}D boundaries{ \displaystyle \varphi }\varphi, the function {\displaystyle V}V will be minimized. If {\displaystyle u}u is the minimized function and {\displaystyle v}v is any smooth function that vanishes on the boundary of {\displaystyle D,}D, then {\displaystyle V[u+\varepsilon v]}V[ u+\varepsilon v] must disappear:
{\displaystyle \left.{\frac {d}{d\varepsilon }}V[u+\varepsilon v]\right| {\varepsilon =0}=\ iint {D}\nabla u\cdot \nabla v ,dx,dy=0.}{\displaystyle \left.{\frac {d}{d\varepsilon }}V[u+\varepsilon v]\right| {\varepsilon =0}= \ iint _{ D}\nabla u\cdot \nabla v,dx,dy=0.}
Assuming u has two derivatives, we can apply the divergence theorem to get
{\displaystyle \iint _{D}\nabla \cdot (v\nabla u),dx,dy=\iint _{D}\nabla u\cdot \nabla v+v\nabla \cdot \nabla u\ ,dx,dy=\int _{C}v{\frac {\partial u}{\partial n}},ds,}{\displaystyle \iint _{D}\nabla \cdot (v\nabla u ),dx,dy=\iint _{D}\nabla u\cdot \nabla v+v\nabla \cdot \nabla u,dx,dy=\int _{C}v{\frac {\部分 u}{\partial n}},ds,}
其中 {\displaystyle C}C 是 {\displaystyle D,}D 的边界，{\displaystyle s}s 是沿 {\displaystyle C}C 和 {\displaystyle \partial u/\partial n}{\displaystyle \ partial u/\partial n} 是 {\displaystyle u}u 在 {\displaystyle C.}{\displaystyle C.} 上的正态导数。因为 {\displaystyle v}v 在 {\displaystyle C}C 上消失，第一个变体消失，结果是
{\displaystyle \iint _{D}v\nabla \cdot \nabla u,dx,dy=0}{\displaystyle \iint _{D}v\nabla \cdot \nabla u,dx,dy= 0}
对于在 {\displaystyle D.}D 的边界上消失的所有平滑函数 v。一维积分情况的证明可以适用于这种情况，以表明
{\displaystyle \nabla \cdot \nabla u=0}{\displaystyle \nabla \cdot \nabla u=0}
in {\displaystyle D.}D.
The difficulty of this reasoning lies in the assumption that the function u to be minimized must have two Derivative. According to Riemann, the existence of a smooth minimization function is guaranteed by a connection to the physical problem: Branes indeed assume configurations with minimal potential energy. Riemann named this idea the Dirichlet principle, in honor of his teacher Peter Gustave Lejeune Dirichlet. However, Weierstrass gives an example of a variational problem with no solution: minimizing

{\displaystyle W[\varphi ]=\int _{-1}^{1}(x\varphi ')^{2},dx}{\displaystyle W[\varphi ]=\int _{-1} ^ {1}(x\varphi ')^{2},dx}
satisfies all {\displaystyle \varphi (-1)=-1}\varphi (-1)=-1 and {\displaystyle \varphi (1) =1.}{\displaystyle \varphi }\varphi (1)=1 in the function of \varphi. {\displaystyle W}W can be made arbitrarily small by choosing a piecewise linear function that transforms between -1 and 1 in a small neighborhood of the origin. However, there is no function such that {\displaystyle W=0.}{\displaystyle W=0.}[j] conclusively proves that Dirichlet's principle works, but it requires a complex application of the theory of regularity of the elliptic part Differential equations; see Jost and Li-Jost (1998).

Generalized to other boundary value problems
A more general expression for the potential energy of the membrane is

{\displaystyle V[\varphi ]=\iint _{D}\left[{\frac {1}{2}}\nabla \varphi \cdot \nabla \varphi +f(x,y)\varphi \right] ,dx,dy,+\int _{C}\left[{\frac {1}{2}}\sigma (s)\varphi ^{2}+g(s)\varphi \right]\ ,ds. }{\displaystyle V[\varphi ]=\iint _{D}\left[{\frac {1}{2}}\nabla \varphi \cdot \nabla \varphi +f(x,y)\ varphi \right ],dx,dy,+\int _{C}\left[{\frac {1}{2}}\sigma (s)\varphi ^{2}+g(s)\varphi \right],ds. }
这对应于 {\displaystyle D,}D 中的外力密度 {\displaystyle f(x,y)}f(x,y)，边界上的外力 {\displaystyle g(s)}g(s) {\displaystyle C,}{\displaystyle C,} 和模量 {\displaystyle \sigma (s)}{\displaystyle \sigma (s)} 作用于 {\displaystyle C.}{\displaystyle C.}将势能最小化且对其边界值没有限制的函数将用 {\displaystyle u.}u 表示。假设 {\displaystyle f}f 和 {\displaystyle g}g 是连续的，正则性理论意味着最小化函数 {\displaystyle u}u 将有两个导数。在采用第一种变化时，不需要对增量 {\displaystyle v.}v 施加边界条件。 {\displaystyle V[u+\varepsilon v]}V[u+\varepsilon v] 的第一个变体由下式给出
{\displaystyle \iint _{D}\left[\nabla u\cdot \nabla v+fv\right],dx,dy+\int _{C}\left[\sigma uv+gv\right], ds=0.}{\displaystyle \iint _{D}\left[\nabla u\cdot \nabla v+fv\right],dx,dy+\int _{C}\left[\sigma uv+gv \right],ds=0.}
如果我们应用散度定理，结果是
{\displaystyle \iint _{D}\left[-v\nabla \cdot \nabla u+vf\right],dx,dy+\int _{C}v\left[{\frac {\partial u} {\ partial n}}+\sigma u+g\right],ds=0.}{\displaystyle \iint _{D}\left[-v\nabla \cdot \nabla u+vf\right], dx,dy+\ int _{C}v\left[{\frac {\partial u}{\partial n}}+\sigma u+g\right],ds=0.}If we first in {\displaystyle C,}{
\ displaystyle C,} set {\displaystyle v=0}v=0, then the boundary integral disappears, and we conclude
{\displaystyle -\nabla \cdot \nabla u+f=0}{\displaystyle -\nabla \cdot \nabla u+f=0}
in {\displaystyle D.}D. Then if we allow {\displaystyle v}v to assume arbitrary boundary values, this means that {\displaystyle u}u must satisfy the boundary condition
{\displaystyle {\frac {\partial u}{\partial n}}+\sigma u+g=0,}{\displaystyle {\frac {\partial u}{\partial n}}+\sigma u+g =0,}
In {\displaystyle C.}{\displaystyle C.} this boundary condition is a consequence of the minimization property of {\displaystyle u}u: it is not imposed in advance. Such conditions are called natural boundary conditions.
If {\displaystyle \sigma }\sigma also disappears on {\displaystyle C.}{\displaystyle C}, the above reasoning is invalid. In this case, we can allow the trial function {\displaystyle \varphi \equiv c }{\displaystyle \varphi \equiv c,} where {\displaystyle c}c is a constant. For a trial function like this,

{\displaystyle V[c]=c\left[\iint _{D}f,dx,dy+\int _{C}g,ds\right].}{\displaystyle V[c]=c\left[\ iint _{D}f,dx,dy+\int _{C}g,ds\right].}
By appropriate choice of {\displaystyle c,}c, {\displaystyle V}V can assume any value unless the quantity inside the brackets vanishes. Therefore, the variational problem is meaningless unless
{\displaystyle \iint _{D}f,dx,dy+\int _{C}g,ds=0.}{\displaystyle \iint _{D }f,dx,dy+\int _{ C}g,ds=0.}
This situation means that the net external forces on the system are in equilibrium. If these forces are in equilibrium, the variational problem has a solution, but it is not unique because arbitrary constants can be added. More details and examples are in Courant and Hilbert (1953).

12. Eigenvalue problem

Both one-dimensional and multidimensional eigenvalue problems can be formulated as variational problems.

Sturm-Liouville Problem
See: Sturm-Liouville Theory
The Sturm-Liouville eigenvalue problem involves general quadratic forms

{\displaystyle Q[\varphi ]=\int {x {1}}^{x_{2}}\left[p(x)\varphi '(x)^{2}+q(x)\varphi ( x )^{2}\right],dx,}{\displaystyle Q[\varphi ]=\int {x {1}}^{x_{2}}\left[p(x)\varphi '( x)^ {2}+q(x)\varphi (x)^{2}\right],dx,}
where {\displaystyle \varphi }\varphi is limited to functions satisfying the boundary conditions
{\displaystyle \varphi (x_{1} )=0,\quad \varphi (x_{2})=0.}{\displaystyle \varphi (x_{1})=0,\quad \varphi (x_{2 })=0.}let{\
displaystyle R}R is the normalized integral
{\displaystyle R[\varphi ]=\int {x {1}}^{x_{2}}r(x)\varphi (x)^{2},dx.}{ \displaystyle R[\varphi ]=\int {x {1}}^{x_{2}}r(x)\varphi (x)^{2},dx.}
The functions {\displaystyle p(x)}p(x) and {\displaystyle r(x)}r(x) need to be everywhere positive and away from zero. The main variational problem is to minimize the ratio {\displaystyle Q/R}{\displaystyle Q/R} of all {\displaystyle \varphi }\varphi satisfying the endpoint conditions. As shown below, the Euler-Lagrange equation that minimizes {\displaystyle u}u is
{\displaystyle -(pu')'+qu-\lambda ru=0,}{\displaystyle -(pu')' +qu-\lambda ru=0,}
where {\displaystyle \lambda }\lambda is the quotient
{\displaystyle \lambda ={\frac {Q[u]}{R[u]}}.}{\displaystyle \lambda ={\frac {Q[u]}{R[u]}}.}
It can be shown (see Gelfand and Fomin 1963) that minimizing {\displaystyle u}u has two derivatives and satisfies the Euler-Lagrange equation . The associated {\displaystyle \lambda }\lambda will be represented as {\displaystyle \lambda {1}}\lambda {1}; it is the lowest eigenvalue of this equation and boundary conditions. The associated minimization function will be denoted as {\displaystyle u {1}(x).}{\displaystyle u {1}(x). } This variational characterization of eigenvalues ​​leads to the Rayleigh-Ritz method: choose an approximate {\displaystyle u}u as a linear combination of basis functions (such as trigonometric functions), and perform a finite-dimensional minimum of change. This method is often surprisingly accurate.
下一个最小的特征值和特征函数可以通过在附加约束下最小化 {\displaystyle Q}Q

{\displaystyle \int {x{1}}^{x_{2}}r(x)u_{1}(x)\varphi (x),dx=0.}{\displaystyle \int {x {1}}^{x_{2}}r(x)u_{1}(x)\varphi (x),dx=0.}
可以扩展此过程以获得问题的特征值和特征函数的完整序列。
变分问题也适用于更一般的边界条件。除了要求 {\displaystyle \varphi }\varphi 在端点处消失，我们可以不在端点处施加任何条件，并设置

{\displaystyle Q[\varphi ]=\int {x{1}}^{x_{2}}\left[p(x)\varphi ‘(x)^{2}+q(x)\varphi ( x)^{2}\right],dx+a_{1}\varphi (x_{1})^{2}+a_{2}\varphi (x_{2})^{2},}{\显示样式 Q[\varphi ]=\int {x{1}}^{x_{2}}\left[p(x)\varphi ‘(x)^{2}+q(x)\varphi (x) ^{2}\right],dx+a_{1}\varphi (x_{1})^{2}+a_{2}\varphi (x_{2})^{2},}
其中 {\displaystyle a_{1}}a_{1} 和 {\displaystyle a_{2}}a_{2} 是任意的。如果我们设置 {\displaystyle \varphi =u+\varepsilon v}{\displaystyle \varphi =u+\varepsilon v}，比率 {\displaystyle Q/R}{\displaystyle Q/R} 的第一个变化是
{\displaystyle V_{1}={\frac {2}{R[u]}}\left(\int {x{1}}^{x_{2}}\left[p(x)u’( x)v’(x)+q(x)u(x)v(x)-\lambda r(x)u(x)v(x)\right],dx+a_{1}u(x_{ 1})v(x_{1})+a_{2}u(x_{2})v(x_{2})\right),}{\displaystyle V_{1}={\frac {2}{R [u]}}\left(\int {x {1}}^{x_{2}}\left[p(x)u'(x)v'(x)+q(x)u(x) v(x)-\lambda r(x) u(x)v(x)\right],dx+a_{1}u(x_{1})v(x_{1})+a_{2}u(x_{2})v(x_{2} )\right),}
where λ is given by the ratio {\displaystyle Q[u]/R[u]}{\displaystyle Q[u]/R[u]}, as described earlier. After division integration,
{\displaystyle {\frac {R[u]}{2}}V_{1}=\int {x {1}}^{x_{2}}v(x)\left[-( pu')' +qu-\lambda ru\right],dx+v(x_{1})[-p(x_{1})u'(x_{1})+a_{1}u(x_{1 })] +v(x_{2})[p(x_{2})u'(x_{2})+a_{2}u(x_{2})].}{\displaystyle {\frac {R [u] }{2}}V_{1}=\int {x {1}}^{x_{2}}v(x)\left[-(pu')'+qu-\lambda ru\right] , dx+v(x_{1})[-p(x_{1})u'(x_{1})+a_{1}u(x_{1})]+v(x_{2})[p ( x_{2})u'(x_{2})+a_{2}u(x_{2})].}
If we first ask {\displaystyle v}v to disappear at the endpoints, the first variant for All such {\displaystyle v}v will disappear only if
{\displaystyle -(pu’)‘+qu-\lambda ru=0\quad {\hbox{for}}\quad x_{1}<x<x_{2}.}{\displaystyle -(pu’)’ +qu-\lambda ru=0\quad {\hbox{for}}\quad x_{1}<x<x_{2}.}
如果 {\displaystyle u}u 满足这个条件，那么对于任意 {\displaystyle v}v，第一个变化将消失，仅当
{\displaystyle -p(x_{1})u’(x_{1})+a_{1}u(x_{1})=0,\quad {\hbox{and}}\quad p(x_{2 })u’(x_{2})+a_{2}u(x_{2})=0.}{\displaystyle -p(x_{1})u’(x_{1})+a_{1} u(x_{1})=0,\quad {\hbox{and}}\quad p(x_{2})u’(x_{2})+a_{2}u(x_{2})=0 .}
后面的这些条件是这个问题的自然边界条件，因为它们不是为了最小化而强加于试验函数，而是最小化的结果。

多维特征值问题
高维特征值问题的定义与一维情况类似。例如，给定一个域 {\displaystyle D}D 在三个维度上具有边界 {\displaystyle B}B，我们可以定义

{\displaystyle Q[\varphi ]=\iiint _{D}p(X)\nabla \varphi \cdot \nabla \varphi +q(X)\varphi ^{2},dx,dy,dz+\ iint _{B}\sigma (S)\varphi ^{2},dS,}{\displaystyle Q[\varphi ]=\iiint _{D}p(X)\nabla \varphi \cdot \nabla \varphi +q(X)\varphi ^{2},dx,dy,dz+\iint _{B}\sigma (S)\varphi ^{2},dS,}
和
{\displaystyle R[\varphi ]=\iiint _{D}r(X)\varphi (X)^{2},dx,dy,dz.}{\displaystyle R[\varphi ]=\iiint _{D}r(X)\varphi (X)^{2},dx,dy,dz.}
令 {\displaystyle u}u 是使商 {\displaystyle Q[\varphi ]/R[\varphi ],}{\displaystyle Q[\varphi ]/R[\varphi ],} 最小化的函数，没有规定条件在边界 {\displaystyle B.}B. {\displaystyle u}u 满足的欧拉-拉格朗日方程为
{\displaystyle -\nabla \cdot (p(X)\nabla u)+q(x)u-\lambda r(x)u=0,}{\displaystyle -\nabla \cdot (p(X)\nabla u)+q(x)u-\lambda r(x)u=0,}
在哪里
{\displaystyle \lambda ={\frac {Q[u]}{R[u]}}.}{\displaystyle \lambda ={\frac {Q[u]}{R[u]}}.}
minimizes {\displaystyle u}u must also satisfy the natural boundary condition
{\displaystyle p(S){\frac {\partial u}{\partial n}}+\sigma (S)u=0,}{\displaystyle p(S ){\frac {\partial u}{\partial n}}+\sigma (S)u=0,}
on the boundary {\displaystyle B.}B. This result depends on the regularity theory of elliptic partial differential equations; See Jost and Li-Jost (1998). Many extensions, including completeness results, asymptotic properties of eigenvalues, and results on eigenfunction nodes are in Courant and Hilbert (1953).

13. Specific applications

Optics
Fermat's principle states that the path of light (locally) minimizes the optical length between its endpoints. If the {\displaystyle x}x coordinate is chosen as the parameter along the path, and along the path {\displaystyle y=f(x)}y=f(x), the optical length is given by

{\displaystyle A[f]=\int {x {0}}^{x_{1}}n(x,f(x)){\sqrt {1+f'(x)^{2}}} dx ,}{\displaystyle A[f]=\int {x {0}}^{x_{1}}n(x,f(x)){\sqrt {1+f'(x)^{2 }} }dx,}
where the refractive index {\displaystyle n(x,y)}n(x,y) depends on the material. If we try {\displaystyle f(x)=f_{0}(x)+\varepsilon f_{1}(x)}{\displaystyle f(x)=f_{0}(x)+\varepsilon f_{1 }(x)} Then the first variant of {\displaystyle A}A (the derivative of {\displaystyle A}A with respect to ε) is {
\displaystyle \delta A[f_{0},f_{1}]=\ int {x {0}}^{x_{1}}\left[{\frac {n(x,f_{0}) f_{0}'(x)f_{1}'(x)}{\sqrt {1+f_{0}'(x)^{2}}}}+n_{y}(x,f_{0} )f_{1}{\sqrt {1+f_{0}'(x)^ {2}}}\right]dx.}{\displaystyle \delta A[f_{0},f_{1}]=\int {x{0}}^{x_{1}}\left[{\frac {n(x,f_{0})f_{0}'(x)f_{1}'(x)}{\ sqrt {1+ f_{0}'(x)^{2}}}}+n_{y}(x,f_{0})f_{1}{\sqrt {1+f_{0}'(x) ^{2} }}\right]dx.}
After partial integration of the first term in brackets, we get the Euler-Lagrange equation

{\displaystyle -{\frac {d}{dx}}\left[{\frac {n(x,f_{0})f_{0}'}{\sqrt {1+f_{0}'^{2 }}}}\right]+n_{y}(x,f_{0}){\sqrt {1+f_{0}'(x)^{2}}}=0.}{\displaystyle -{\ frac {d}{dx}}\left[{\frac {n(x,f_{0})f_{0}'}{\sqrt {1+f_{0}'^{2}}}}\right ]+n_{y}(x,f_{0}){\sqrt {1+f_{0}'(x)^{2}}}=0.} Rays
can be determined by integrating this equation. This formalism is used in the context of Lagrange optics and Hamiltonian optics.

Snell's Law
When light enters or exits a lens, there is a discontinuity in the index of refraction. let

{\displaystyle n(x,y)={\begin{cases}n_{(-)}&{\text{if}}\quad x<0,\n_{(+)}&{\text{if }}\quad x>0,\end{cases}}}{\displaystyle n(x,y)={\begin{cases}n_{(-)}&{\text{if}}\quad x<0 ,\n_{(+)}&{\text{if}}\quad x>0,\end{cases}}}
其中 {\displaystyle n_{(-)}}{\displaystyle n_{(-)}} 和 {\displaystyle n_{(+)}}{\displaystyle n_{(+)}} 是常数。那么欧拉-拉格朗日方程在 {\displaystyle x<0}x<0 或 {\displaystyle x>0,}{\displaystyle x>0,} 的区域和以前一样成立，实际上路径在那里是一条直线，因为折射率是恒定的。在 {\displaystyle x=0,}{\displaystyle x=0,} {\displaystyle f}f 必须是连续的，但 {\displaystyle f’}f’ 可能是不连续的。在单独区域中按部分积分并使用欧拉-拉格朗日方程后，第一个变化形式为
{\displaystyle \delta A[f_{0},f_{1}]=f_{1}(0)\left[n_{(-)}{\frac {f_{0}‘(0^{-}) }{\sqrt {1+f_{0}’(0^{-}){2}}}}-n_{(+)}{\frac {f_{0}‘(0^{+})} {\sqrt {1+f_{0}’(0^{+}){2}}}}\right].}{\displaystyle \delta A[f_{0},f_{1}]=f_{ 1}(0)\left[n_{(-)}{\frac {f_ {0}'(0^{-})}{\sqrt {1+f_{0}'(0 ^{-}) {2}}}}-n_{(+)}{\frac {f_{0} '(0^{+})}{\sqrt {1+f_{0}'(0 ^{+}) { 2}}}}\right]. }
multiplied by {\displaystyle n_{(-)}}{\displaystyle n_{(-)}} is the sine of the angle between the incident ray and the {\displaystyle x}x-axis, multiplied by {\displaystyle n_ {( +)}}{\displaystyle n_{(+)}} is the sine of the angle of the refracted ray with the {\displaystyle x}x-axis. Snell's law of refraction requires these terms to be equal. As this calculation shows, Snell's law is equivalent to the disappearance of the first change in optical path length.

It is convenient to use the vector notation for Fermat's principle in three dimensions : let {\displaystyle X=(x_{1},x_{2},x_{3}),}{\displaystyle X=(x_{1}, x_{2},x_{ 3}),} Let {\displaystyle t}t be the parameter, let {\displaystyle X(t)}X(t) be the curve {\displaystyle C,}{\displaystyle C,} { \displaystyle {\dot {X}}(t)}{\dot {X}}(t) is its tangent vector. The optical length of the curve is given by

{\displaystyle A[C]=\int {t {0}}^{t_{1}}n(X){\sqrt { { \dot {X}}\cdot {\dot {X}}}} , dt.}{\displaystyle A[C]=\int {t {0}}^{t_{1}}n(X){\sqrt { {\dot {X}}\cdot {\dot { X}} }},dt.}
Note that this integral is invariant to changes in the parameter representation of {\displaystyle C.}{\displaystyle C.}. The Euler-Lagrange equation of the minimized curve has a symmetric form

{\displaystyle {\frac {d}{dt}}P={\sqrt { {\dot {X}}\cdot {\dot {X}}}},\nabla n,}{\displaystyle {\frac { d}{dt}}P={\sqrt { {\dot {X}}\cdot {\dot {X}}}},\nabla n,}
where
{\displaystyle P={\frac {n(X ){\dot {X}}}{\sqrt { {\dot {X}}\cdot {\dot {X}}}}}.}{\displaystyle P ={\frac {n(X){\dot {X}}}{\sqrt { {\dot {X}}\cdot {\dot {X}}}}}. }
By definition, {\displaystyle P}P satisfies

{\displaystyle P\cdot P=n(X)^{2}.}{\displaystyle P\cdot P=n(X)^{2}.}
因此，积分也可以写成

{\displaystyle A[C]=\int {t{0}}^{t_{1}}P\cdot {\dot {X}},dt.}{\displaystyle A[C]=\int _ {t_{0}}^{t_{1}}P\cdot {\dot {X}},dt.}
这种形式表明，如果我们可以找到一个函数 {\displaystyle \psi }\psi，其梯度由 {\displaystyle P,}P 给出，那么积分 {\displaystyle A}A 由 {\displaystyle \ psi }\psi 在积分区间的端点。因此，研究使积分平稳的曲线问题可以与研究 {\displaystyle \psi .}\psi 的水平面有关。为了找到这样的函数，我们求助于波动方程，它支配光的传播。这种形式主义在拉格朗日光学和哈密顿光学的背景下使用。

与波动方程的联系
非均匀介质的波动方程为

{\displaystyle u_{tt}=c^{2}\nabla \cdot \nabla u,}{\displaystyle u_{tt}=c^{2}\nabla \cdot \nabla u,}where{\displaystyle c
} c is speed, usually depends on {\displaystyle X.}X. The wavefronts of light are the characteristic surfaces of this partial differential equation: they satisfy
{\displaystyle \varphi _{t} ^{2}=c(X) {2},\nabla \varphi \cdot \nabla \varphi .}{\ displaystyle \varphi _{t}^{2}= c(X)^{2},\nabla \varphi \cdot \nabla \varphi .}
We may look for solutions in the form

{\displaystyle \varphi (t,X)=t-\psi (X).}{\displaystyle \varphi (t,X)=t-\psi (X).}In this case, {\displaystyle
\ psi }\psi satisfies

{\displaystyle \nabla \psi \cdot \nabla \psi =n^{2},}{\displaystyle \nabla \psi \cdot \nabla \psi =n^{2},}where {\displaystyle n=1
/ c.}n=1/c. According to the theory of first-order partial differential equations, if {\displaystyle P=\nabla \psi ,}{\displaystyle P=\nabla \psi ,} then {\displaystyle P}P satisfies {\displaystyle {\frac {dP}{
ds }}=n,\nabla n,}{\displaystyle {\frac {dP}{ds}}=n,\nabla n,}
along the system of curves (rays) given by
{\displaystyle {\frac {dX}{ds}}=P.}{\displaystyle {\frac {dX}{ds}}=P.}
If we identify, these equations for solving first-order partial differential equations are similar to Euler-Lager Langer's equation is the same

{\displaystyle {\frac {ds}{dt}}={\frac {\sqrt { {\dot {X}}\cdot {\dot {X}}}}{n}}.}{\displaystyle {\ frac {ds}{dt}}={\frac {\sqrt { {\dot {X}}\cdot {\dot {X}}}}{n}}。}
我们得出结论，函数 {\displaystyle \psi }\psi 是作为上端点函数的最小化积分 {\displaystyle A}A 的值。即，当构建最小化曲线族时，光程的值满足与波动方程对应的特征方程。因此，求解相关的一阶偏微分方程等效于找到变分问题的解族。这是汉密尔顿-雅可比理论的基本内容，适用于更一般的变分问题。

力学

在经典力学中，作用 {\displaystyle S,}S 被定义为拉格朗日函数 {\displaystyle L.}L 的时间积分。拉格朗日是能量的差异，

{\displaystyle L=TU,}{\displaystyle L=TU,}
where {\displaystyle T}T is the kinetic energy of the mechanical system and {\displaystyle U}U is its potential energy. Hamilton's principle (or principle of action) states that the motion of a conservative complete (integrable constrained) mechanical system is such that the action integral
{\displaystyle S=\int {t {0}}^{t_{1}}L(x,{\ dot {x}},t),dt}{\displaystyle S=\int {t { 0}}^{t_{1}}L(x,{\dot {x}},t),dt}
for the path The variation of {\displaystyle x(t).}{\displaystyle x(t).} is static. The Euler-Lagrangian equations for this system are called the Lagrange equations:
{\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {x}} }}={\frac {\partial L}{\partial x}},} {\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {x}} }}={\frac {\partial L}{\partial x}},}
They are equivalent to Newton's equations of motion (for such systems).
The conjugate momentum {\displaystyle P}P is defined as

{\displaystyle p={\frac {\partial L}{\partial {\dot {x}}}}.}{\displaystyle p={\frac {\partial L}{\partial {\dot {x}} }}. }
For example, if
{\displaystyle T={\frac {1}{2}}m{\dot {x}}^{2},}{\displaystyle T={\frac {1}{2}}m{ \dot {x }}^{2},}
Then
{\displaystyle p=m{\dot {x}}.}{\displaystyle p=m{\dot {x}}.}
If by Lagrangian{ \displaystyle L}L to Hamiltonian {\displaystyle H The Legendre transformation introduces conjugate momentum instead of {\displaystyle {\dot {x}}}{\dot {x}} }H is defined as {\displaystyle H
( x,p,t)=p,{\dot {x}}-L(x,{\dot {x}},t).}{\displaystyle H(x,p,t) =p,{\dot {x}}-L(x,{\dot {x}},t). }
The Hamiltonian is the total energy of the system: {\displaystyle H=T+U.}{\displaystyle H=T+U.} An analogy with Fermat's principle shows that the solution of the Lagrange equation (particle trajectories ) can be used to describe the horizontal plane of a function of {\displaystyle X.}X in terms of This function is the solution of the Hamilton-Jacobi equation:
{\displaystyle {\frac {\partial \psi }{\partial t}}+H\left(x,{\frac {\partial \psi }{\partial x}},t\right)=0.}{ \displaystyle {\frac {\partial \psi }{\partial t}}+H\left(x,{\frac {\partial \psi }{\partial x}},t\right)=0.}

14. Variations and sufficient condition for a minimum

Variational methods focus on the variation of a functional, which is the small change in the value of the functional due to small changes in the functions that are its parameters. The first variation is defined as the linear part of the change in the functional, and the second variation is defined as the quadratic part.

For example, if $J\left[y\right]$ is a function$y=y\left(x\right)$ as a parameter of the functional, and its parameters from$y$到$y+h$ with a small change, where$h=h\left(x\right)$ is related to$y$ is in the same function space, the corresponding change of the functional is:

$$\Delta J[h]=J[y+h]-J\left[y\right]$$

Functional $J\left[y\right]$ is said to be differentiable if:

$$\Delta J\left[h\right]=\phi \left[h\right]+\epsilon \Vert h\Vert$$

where $\phi \left[h\right]$ is a linear functional,$\Vert h\Vert$ ishhThe norm of$h$$\Vert h\Vert \to 0$，$e\to 0$ . Linear functional$\phi [h]$ 是 J [ y ] J[y] The first-order variation of$J$$[$$y$$]\; ,\; and\; expressed\; as:$

$$\delta J[h]=\phi \left[h\right]$$

Functional $J\left[y\right]$ is said to be quadratically differentiable if:

$$\Delta J[h]={Phi}_1[h]+{Phi}_2\left[h\right]+\epsilon \Vert h{\Vert }^2$$

其中 ${\phi }_1[h]$ 是线性泛函（一阶变分），${\phi }_2[h]$ 是二次泛函，随着 $\Vert h\Vert \to 0$，$\epsilon \to 0$。${\phi }_2[h]$ 是 $J[y]$ 的二次泛函，并表示为：

$${\delta }^{2}J[h]={\phi }_2[h]$$

二阶变分 ${\delta }^{2}J\left[h\right]$is considered to be stongly positive if for all$h$ and some constant$k>0$：

$$d^{2}J\left[h\right]\ge k\Vert h{\Vert }^2$$

Using the above definitions, especially the definitions of first-order variation, second-order variation, and constant positive, the following sufficient conditions for minimum functionals can be stated.

A sufficient condition for a minimum value :

if at $y=\hat{y}$, the first-order variation $\delta J[h]=0$，在$y=\hat{y}$, the second-order variation $d^{2}J\left[h\right]$is always positive, then the functional$J\left[y\right]$在$y=\hat{y}$has a minimum value.

15. Fundamental lemma of calculus of variations

In mathematics, especially in the calculus of variations, the function ffThe variation of$f$$\delta f$ can be concentrated on arbitrarily small intervals, but not on a single point$.$Therefore, the necessary condition for extrema (the function derivative is zero) occurs with any function$\delta f$ integrated weak formula (weak formulation) (variational form). The fundamental lemma of calculus of variations is often used to transform this weak formulation into a strong formulation (differential equation) without integration with arbitrary functions. Prove that this possibility is usually used to choose$\delta f$，$\delta f$ is centered so that$f$ holds the sign (positive or negative) on the interval. There are several versions of the lemma. The most basic version is easy to express and prove.

15.1 Basic form

If the open interval $(a,$The continuous function ffon$b$$)$$f$ for$(a,b)$ smooth functionhhfor all compactlysupported smooth functions$h$ , satisfying the equation:

$${\int }_a^{b}f(x)h(x)\mathrm{d}x=0$$

then $f$ is zero.

"Smooth" here can be interpreted as "infinitely differentiable", but is usually interpreted as "twice continuously differentiable" or "continuously differentiable", or even just "continuously", since these weaker statements are less effective for a given task Enough is enough. "Compactly supported" means "for some $c$，$d$ 使得 $a<c<d<b$"；但通常一个较弱的陈述就足够了，假设只有 $h$（或 $h$ 及其许多衍生物）在端点 $a$，$b$；在这种情况下使用闭区间 $[a,b]$。

15.2 两个给定函数的形式

如果区间 $(a,b)$ 上的一对连续函数 $f$，$g$ 对于 $(a,b)$ 上的所有紧支持的平滑函数 $h$，满足等式：

$${\int }_a^b(f(x)h(x)+g(x)h^{\prime }(x))\mathrm{d}x=0$$

则 $g$ 是可微的，并且处处 $g^{\prime }=f$。

$g=0$ 时的特殊情况只是基本形式。

另外当 $f=0$ 时的特殊情况（通常足够）：对于 (a,b) 上的所有平滑函数 $h$ 满足 $h(a)=h(b)=0$ if the interval$(a,$The continuous function ggon$b$$)$$g$ satisfies the equation:

$${\int }_a^{b}g(x)h^{\prime }(x)\mathrm{d}x=0$$

then $g$ is a constant.

Furthermore, if assuming $g$ is continuously differentiable, then integration by parts reduces both formulations to the basic form; this example is given by Joseph-Louis Lagrange, andggThe proof of the differentiability of$g\; is\; given\; by\; Paul\; du\; Bois-Reymond.$

15.3 Forms of discontinuous functions

Given a function $(f,g)$ can be discontinuous as long as they are locally integrable (over a given interval). In this case, the Lebesgue integral means that the conclusion is almost everywhere (thus, at all continuity points),$The\; differentiability\; of\; g$ is interpreted as local absolute continuity (rather than continuous differentiability). Sometimes a given function is assumed to be piecewise continuous, in which case the Riemann integral suffices, stating the conclusion everywhere except for a finite set of discontinuous points.

15.4 Forms under higher derivatives

If a continuous function tuple $f_0,f_1,\cdots ,f_n$For $(a,b)$ on all tightly supported smoothing functions$h$ , in the interval$(a,b)$ satisfies the equation:

$${\int }_a^b(f_0(x)h(x)+f_1(x)h^{\prime }(x)+\cdots +f_n(x)h^{(n)}(x))\mathrm{d}x=0$$

Then there are continuous differentiable functions $u_0,u_1,\dots ,u_{n-1}$, at $(a,b)$ above such that:

$$\begin{array}{rl}{f}_0& =u_0^{\prime },\\ f_1& =u_0+u_1^{\prime },\\ f_2& =u_1+u_2^{\prime }\\ ⋮& \\ f_{n-1}& =u_{n-2}+u_{n-1}^{\prime },\\ f_n& =u_{n-1}\end{array}$$

This necessary condition is also sufficient, since the integrand becomes:

$$(u_{0}h{)}^{\prime }+(u_1{h}^{\prime }{)}^{\prime }+\cdots +(u_{n-1}{h}^{(n-1)}{)}^{\prime }$$

$n=The\; case\; of\; 1$ is just two versions of the given function, since$f=f_0=u_0^{\prime }$ 和 $f_1=u_0$, therefore, $f_0-f_1^{\prime }=0$。

Instead, $n=2$ does not lead to the relation$f_0-f_1^{\prime }+f_2^{\prime \prime }=0$ , because the function$f_2=u_1$There is no need to be differentiable twice. Sufficient condition $f_0-f_1^{\prime }+f_2^{\prime \prime }=0$ is unnecessary. Conversely, for$n=2$ The sufficient and necessary condition can be written as$f_0-(f_1-f_2^{\prime }{)}^{\prime }=0$ , for$n=3$，$f_0-(f_1-(f_2-f_3^{\prime }{)}^{\prime }{)}^{\prime }=0$ , and so on; in general, parentheses cannot be opened due to non-differentiability.

15.5 Vector-valued functions

Generalization to vector-valued functions $(a,b)\to {\mathbb{R}}^d$ is simple; apply the result of a scalar function to each coordinate separately, or handle the vector-valued case from scratch.

15.6 Multivariable functions

If the open set $Oh\subset R^{}$continuous multivariate functionff${}^{on\; d}$$f$ forΩ \OmegaAll tightly supported smoothing functionshh$on\; \Omega$$h$，满足等式：

$${\int }_{\Omega }f(x)h(x)\mathrm{d}x=0$$

则 $f$ 完全为零。

与基本形式类似，可以考虑 $\Omega$ 闭包上的连续函数 $f$，假设 $h$ 在 $\Omega$ 的边界上消失（而不是紧支持）。

下面是不连续多变量函数的形式。

令 $\Omega \subset {\mathbb{R}}^d$ 为开集，且 $f\in L^2(\Omega )$ ，对于 $\Omega$ 上的所有紧支持的平滑函数 $h$，满足等式：

$${\int }_{\Omega }f(x)h(x)\mathrm{d}x=0$$

then $f=0$ (in$L^2$ , ie almost everywhere).

15.7 Compact support

• Compact space

In mathematics, especially general topology, compactness is a property that aims to generalize the closure and The notion of a bounded subset, that is, the space does not exclude any "limiting values" of points. For example, the "unclosed" interval $(0,1)$ won't be compact because it excludes$0$ and1 1The "limit value" of$1$$[0,1]$ will be compact. Similarly, the space of rational numbers$Q$ is not compact because it has an infinite number of "holes" corresponding to irrational numbers, and the real number space$R$ is also not compact because it excludes the limit value$\infty$ and$-\infty$ . However, the extended real number line will be compact because it contains two infinities. There are many ways to make this heuristic concept precise. These ways are generally consistent in Euclidean spaces, but may not be equivalent in other topological spaces.

The interval A = (-∞, -2] is not compact according to the compactness criterion for Euclidean spaces stated in the Heine-Borel theorem because it is unbounded. The interval C = (2, 4) is not compact because it is not closed. The interval B = [0, 1] is compact because it is both closed and bounded.

One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point in the space . ( Bolzano-Weierstrass theorem ) states that a subset of a Euclidean space is compact in this order sense if and only if it is closed and bounded. Therefore, if in the closed unit interval $[0,1]$ , some of which will be arbitrarily close to some real number in that space. For example, the sequence$1/2,4/5,1/3,5/6,1/4,6/7,\cdots$ 中的一些数字累积为 $0$（而其他数字累积为 $1$）。同一组点不会累加到开单位区间 (0, 1) 的任何一点上，因此开单位区间不是紧致的。尽管欧几里得空间的子集（子空间）可以是紧致的，但整个空间本身并不紧致，因为它是无界的。例如，考虑 ${\mathbb{R}}^1$，即整个实数线，点的序列 $0,1,2,3,\cdots$ , 没有收敛到任何实数的子序列。

Compactness was formally introduced by Maurice Fréchet in 1906 to generalize the Bolzano-Weierstrass theorem from geometric point spaces to function spaces. The Arzelà-Ascoli theorem and Peano's existence theorem exemplify the application of this notion of compactness to classical analysis. After the initial introduction, various equivalent notions of compactness were developed in general metric spaces, including sequential compactness and limit-point compactness. However, in general topological spaces these notions of compactness are not necessarily equivalent. The most useful concept and standard definition of unqualified term compactness is stated as follows: the existence of finite families of open sets that "cover" the space, that is, every point of the space lies at In a set contained in the open set. This more subtle concept, proposed by Pavel Alexandrov and Pavel Urysohn in 1929, presents compact spaces as generalizations of finite sets. In compact spaces in this sense, information that exists locally (i.e., the neighborhood of each point) can often be pieced together into corresponding statements that exist in the entire space, and many theorems share this property.

In spaces that are compact in this sense, it is often possible to piece together local information, that is, information within the neighborhood of each point, into corresponding statements that run through the entire space, and many theorems have this nature.

The term "compact set" is sometimes used synonymously with a compact space, but usually also refers to a compact subspace of a topological space.

• Support

In mathematics, the real-valued function ffThe support of$f\; is\; the\; subset\; of\; the\; domain\; containing\; elements\; not\; mapped\; to\; zero.$if$The\; domain\; of\; f$ is a topological space, then$The\; support\; of\; f$ is defined as the smallest closed set containing all points not mapped to zero. This concept is widely used in mathematical analysis.

Suppose $f:X\to R$ is a real-valued function whose domain is any set$X$。 f f set-theoretic support for$f$$\mathrm{supp}\left(f\right)$ which is$The\; geometry\; of\; the\; point\; in\; X$ , where the corresponding$f$ is not zero:

supp ⁡ ( f ) = { x ∈ X   :   f ( x ) ≠ 0 } \operatorname {supp} (f)=\{x\in X\,:\,f(x)\neq 0\} supp(f)={ x∈X:f(x)=0}

$The\; support\; of\; f$ is$The\; smallest\; subset\; of\; X$ whose property is$f$ is zero on the complement of the subset. If for all finite points$x\in X$，$f(x)=0$ , then say$f$ has limited support (finite support).

If set $X$ has an additional structure (eg, a topology), then$The\; support\; of\; f$ is defined to be similar to an appropriate type ofXXThe smallest subset of$X$$f$ disappears in its proper sense on its complement. The concept of support is also extended in a natural way beyond$Accessors\; and\; other\; objects\; in\; R\text{'}s$ more general collections, such as measures or distributions.

• Closed support

The most common situation occurs in $X$ is a topological space (e.g. real line or$n-$ dimensional Euclidean space) and$f:X\to R$ is a continuous real (or complex)-valued function. In this case,$The\; support\; of\; f$ is topologically defined asXXA closure of a subset of$X$ ( closure, in topology, a subsetSSThe closure of$S$ is defined by SSAll points in$S\; and\; all\; limit\; points\; of\; S.$$The\; closure\; of\; S$ can be equivalently defined as$The\; union\; of\; S$ and its bounds can also be defined to containSSThe intersection of all closed sets of$S.$Intuitively, closures can be thought of asSSIn or "near"$SS$$All\; points\; in\; S.$ A point in the closure of S isSSThe closure point of$S.$) (taken from$X$ ), where$f$ is not zero, that is:

supp ⁡ ( f ) : = cl ⁡ X ( { x ∈ X   :   f ( x ) ≠ 0 } ) = f − 1 ( { 0 } c ) ‾ \operatorname {supp} (f):=\operatorname {cl} _{X}\left(\{x\in X\,:\,f(x)\neq 0\}\right)={ \overline {f^{-1}\left(\{0\}^{c}\right)}} supp(f):=clX​({ x∈X:f(x)=0})=f−1({ 0}c)​

由于闭集的交集是闭集，$\mathrm{supp}(f)$ 是所有包含 $f$ 集合论支持（set-theoretic）的封闭集的交集。

例如，如果函数 $f:\mathbb{R}\to \mathbb{R}$ 被定义为：

$$f(x)=\{\begin{array}{ll}1-x^2& \text{if }|x|<1\\ 0& \text{if }|x|\ge 1\end{array}$$

then $The\; support\; of\; f$ is the closed interval$[-1,1]$ , because$f$ is in the open interval$(-1,1)$ is non-zero, and the closure of this set is$[-1,1]$。

The notion of closed support is usually applied to continuous functions, but the definition makes sense for arbitrary real or complex-valued functions on topological spaces, and some authors do not require $f:X\to \mathbb{R}$ 或 $f:X\to C$ is continuous.

• Compact support

In the topological space XXFunctions with tight support on$X$ are those whose closed support is XXA function for a compact subset of$X.$If$X$ is a solid line, or$n$ -dimensional Euclidean space, then a function has compact support if and only if it has bounded support (bounded support), because${\mathbb{R}}^{A\; subset\; of\; n}$ is compact if and only if it is closed and bounded.

For example, the function f defined above $f:R\to R$ is a compact support$[-1,1]$ as a continuous function. If$f:R^n\to R$ is a smooth function because$f$ in the open subset${\mathbb{R}}^n\setminus \mathrm{supp}\left(f\right)$ with$0$ same asffThe partial derivatives of all orders of$f$$R^n\setminus \mathrm{supp}\left(f\right)$ is also with$0$ is exactly the same.

The condition for tight support is stronger than the condition for vanishing at infinity. For example, the function $f:R\to R$ is defined as

$$f(x)=\frac{1}{1+x^2}$$

vanishes at infinity because as $|x|\to \infty$，$f(x)\to 0$ , but it's supported by$R$ is not compact.

Real-valued compactly supported smoothing functions on Euclidean spaces are called bump functions. Mollifiers are an important special case of bump functions because they can be used in distribution theory to create sequences of smooth functions that approximate non-smooth (generalized) functions by convolution.

In the good case, functions with tight support are dense in the space of functions vanishing at infinity , but this property requires some technical work to justify for a given example. As an intuition for more complex examples, in a limit language, for any $e>0$ , any function$f$ on the real line$R$ , which can be selected by selectingR \mathbb {R}An appropriate compact subset of$R$$C$ to approximate such that

$$\left|f(x)-I_C(x)f(x)\right|<e$$

For all $x\in X$ , where$I_C$is $C$ indicator function (Indicator function). Every continuous function on a compact topological space has compact support because every closed subset of a compact space is indeed compact.

16. More applications

• Derivation of the shape of the catenary
• The Solution of Newton's Minimum Resistance Problem
• The solution to the brachistochrone problem
• The solution to the tautochrone problem
• Solutions to Isometric Problems
• Calculate geodesics
• Find the minimum surface and solve the plateau problem
• best control
• Analytical mechanics, or reformulations of Newton's laws of motion, most notably Lagrangian and Hamiltonian mechanics;
• Geometric optics, especially Lagrangian and Hamiltonian optics;
• Calculus of Variations (Quantum Mechanics), a method for finding the lowest energy eigenstate or ground state and an approximation of some excited states;
• Variational Bayesian methods, a set of techniques for approximating intractable integrals that arise in Bayesian inference and machine learning;
• Variational methods in general relativity, a series of techniques that use variational methods to solve problems in Einstein's general theory of relativity;
• The finite element method is a variational method for finding numerical solutions of boundary value problems in differential equations;
• Total Variation Denoising, an image processing method for filtering high variance or noisy signals.

Almost all the basic laws of physics and mechanics are stated as the "principle of variational method" that stipulates that the variation of a certain functional should be 0. For this reason, the variational method enables many important physical and technical problems to be solved.

17. Introduction to Functional Analysis

wiki: Functional analysis

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• References:

【Mathematics Encyclopedia】What is variation? How is it different from differential?

wiki: Calculus of variations

wiki: Euler–Lagrange equation

wiki: Fundamental lemma of calculus of variations

wiki:Compact space

wiki: Mollifier

wiki: Bump function

wiki: Characteristic function

wiki: Indicator function

Beltrami identity