Editing Calculus of variations (section)

== Euler–Lagrange equation ==
{{main|Euler–Lagrange equation}}
Finding the extrema of functionals is similar to finding the maxima and minima of functions. The maxima and minima of a function may be located by finding the points where its derivative vanishes (i.e., is equal to zero).  The extrema of functionals may be obtained by finding functions for which the [[functional derivative]] is equal to zero. This leads to solving the associated [[Euler–Lagrange equation]].{{efn|The following derivation of the Euler–Lagrange equation corresponds to the derivation on pp. 184–185 of Courant & Hilbert (1953).<ref>{{cite book |author=Courant, R. |author-link=Richard Courant |author2=Hilbert, D. |author2-link= David Hilbert |title=Methods of Mathematical Physics |volume=I |edition=First English |publisher=Interscience Publishers, Inc. |year=1953 |location=New York |isbn=978-0471504474}}</ref>}}

Consider the functional
<math display="block">J[y] = \int_{x_1}^{x_2} L\left(x,y(x),y'(x)\right)\, dx \, .</math>
where
*<math>x_1, x_2</math> are [[Constant (mathematics)|constants]],
*<math>y(x)</math> is twice continuously differentiable,
*<math>y'(x) = \frac{dy}{dx},</math>
*<math>L\left(x, y(x), y'(x)\right)</math> is twice continuously differentiable with respect to its arguments <math>x, y,</math> and <math>y'.</math>

If the functional <math>J[y]</math> attains a [[local minimum]] at <math>f,</math> and <math>\eta(x)</math> is an arbitrary function that has at least one derivative and vanishes at the endpoints <math>x_1</math> and <math>x_2,</math> then for any number <math>\varepsilon</math> close to 0,
<math display="block">J[f] \le J[f + \varepsilon \eta] \, .</math>

The term <math>\varepsilon \eta</math> is called the '''variation''' of the function <math>f</math> and is denoted by <math>\delta f.</math><ref name='CourHilb1953P184'/>{{efn|Note that <math>\eta(x)</math> and <math>f(x)</math> are evaluated at the {{em|same}} values of <math>x,</math> which is not valid more generally in variational calculus with non-holonomic constraints.}}

Substituting <math>f + \varepsilon \eta</math> for <math>y</math> in the functional <math>J[y],</math> the result is a function of <math>\varepsilon,</math>

<math display="block">\Phi(\varepsilon) = J[f+\varepsilon\eta] \, .</math>
Since the functional <math>J[y]</math> has a minimum for <math>y = f</math> the function <math>\Phi(\varepsilon)</math> has a minimum at <math>\varepsilon = 0</math> and thus,{{efn|The product <math>\varepsilon \Phi'(0)</math> is called the first variation of the functional <math>J</math> and is denoted by <math>\delta J.</math>  Some references define the [[first variation]] differently by leaving out the <math>\varepsilon</math> factor.}}
<math display="block">\Phi'(0) \equiv \left.\frac{d\Phi}{d\varepsilon}\right|_{\varepsilon = 0} = \int_{x_1}^{x_2} \left.\frac{dL}{d\varepsilon}\right|_{\varepsilon = 0} dx = 0 \, .</math>

Taking the [[total derivative]] of <math>L\left[x, y, y'\right],</math> where <math>y = f + \varepsilon \eta</math> and <math>y' = f' + \varepsilon \eta'</math> are considered as functions of <math>\varepsilon</math> rather than <math>x,</math> yields
<math display="block">\frac{dL}{d\varepsilon}=\frac{\partial L}{\partial y}\frac{dy}{d\varepsilon} + \frac{\partial L}{\partial y'}\frac{dy'}{d\varepsilon}</math>
and because <math>\frac{dy}{d \varepsilon} = \eta</math> and <math>\frac{d y'}{d \varepsilon} = \eta',</math>
<math display="block">\frac{dL}{d\varepsilon}=\frac{\partial L}{\partial y}\eta + \frac{\partial L}{\partial y'}\eta'.</math>

Therefore,
<math display="block">\begin{align}
\int_{x_1}^{x_2} \left.\frac{dL}{d\varepsilon}\right|_{\varepsilon = 0} dx
 & = \int_{x_1}^{x_2} \left(\frac{\partial L}{\partial f} \eta + \frac{\partial L}{\partial f'} \eta'\right)\, dx \\
 & = \int_{x_1}^{x_2} \frac{\partial L}{\partial f} \eta \, dx + \left.\frac{\partial L}{\partial f'} \eta \right|_{x_1}^{x_2} - \int_{x_1}^{x_2} \eta \frac{d}{dx}\frac{\partial L}{\partial f'} \, dx \\
 & = \int_{x_1}^{x_2} \left(\frac{\partial L}{\partial f} \eta - \eta \frac{d}{dx}\frac{\partial L}{\partial f'} \right)\, dx\\
\end{align}</math>
where <math>L\left[x, y, y'\right] \to L\left[x, f, f'\right]</math> when <math>\varepsilon = 0</math> and we have used [[integration by parts]] on the second term.  The second term on the second line vanishes because <math>\eta = 0</math> at <math>x_1</math> and <math>x_2</math> by definition.  Also, as previously mentioned the left side of the equation is zero so that
<math display="block">\int_{x_1}^{x_2} \eta (x) \left(\frac{\partial L}{\partial f} - \frac{d}{dx}\frac{\partial L}{\partial f'} \right) \, dx = 0 \, .</math>

According to the [[fundamental lemma of calculus of variations]], the part of the integrand in parentheses is zero, i.e.
<math display="block">\frac{\partial L}{\partial f} -\frac{d}{dx} \frac{\partial L}{\partial f'}=0</math>
which is called the '''Euler–Lagrange equation'''.  The left hand side of this equation is called the [[functional derivative]] of <math>J[f]</math> and is denoted <math>\delta J</math> or <math>\delta f(x).</math>

In general this gives a second-order [[ordinary differential equation]] which can be solved to obtain the extremal function <math>f(x).</math>  The Euler–Lagrange equation is a [[Necessary condition|necessary]], but not [[Sufficient condition|sufficient]], condition for an extremum <math>J[f].</math>  A sufficient condition for a minimum is given in the section [[#Variations and sufficient condition for a minimum|Variations and sufficient condition for a minimum]].

=== Example ===
In order to illustrate this process, consider the problem of finding the extremal function <math>y = f(x),</math> which is the shortest curve that connects two points <math>\left(x_1, y_1\right)</math> and <math>\left(x_2, y_2\right).</math> The [[arc length]] of the curve is given by
<math display="block">A[y] = \int_{x_1}^{x_2} \sqrt{1 + [ y'(x) ]^2} \, dx \, ,</math>
with
<math display="block">y'(x) = \frac{dy}{dx} \, , \ \ y_1=f(x_1) \, , \ \ y_2=f(x_2) \, .</math>
Note that assuming {{mvar|y}} is a function of {{mvar|x}} loses generality; ideally both should be a function of some other parameter. This approach is good solely for instructive purposes.

The Euler–Lagrange equation will now be used to find the extremal function <math>f(x)</math> that minimizes the functional <math>A[y].</math>
<math display="block">\frac{\partial L}{\partial f} -\frac{d}{dx} \frac{\partial L}{\partial f'}=0</math>
with
<math display="block">L = \sqrt{1 + [ f'(x) ]^2} \, .</math>

Since <math>f</math> does not appear explicitly in <math>L,</math> the first term in the Euler–Lagrange equation vanishes for all <math>f(x)</math> and thus,
<math display="block">\frac{d}{dx} \frac{\partial L}{\partial f'} = 0 \, .</math>
Substituting for <math>L</math> and taking the derivative,
<math display="block">\frac{d}{dx} \ \frac{f'(x)} {\sqrt{1 + [f'(x)]^2}} \ = 0 \, .</math>

Thus
<math display="block">\frac{f'(x)}{\sqrt{1+[f'(x)]^2}} = c \, ,</math>
for some constant <math>c.</math> Then
<math display="block">\frac{[f'(x)]^2}{1+[f'(x)]^2} = c^2 \, ,</math>
where
<math display="block">0 \le c^2<1.</math>
Solving, we get
<math display="block">[f'(x)]^2=\frac{c^2}{1-c^2}</math>
which implies that
<math display="block">f'(x)=m</math>
is a constant and therefore that the shortest curve that connects two points <math>\left(x_1, y_1\right)</math> and <math>\left(x_2, y_2\right)</math> is
<math display="block">f(x) = m x + b \qquad \text{with} \ \ m = \frac{y_2 - y_1}{x_2 - x_1} \quad \text{and} \quad b = \frac{x_2 y_1 - x_1 y_2}{x_2 - x_1}</math>
and we have thus found the extremal function <math>f(x)</math> that minimizes the functional <math>A[y]</math> so that <math>A[f]</math> is a minimum. The equation for a straight line is <math>y = mx+b.</math> In other words, the shortest distance between two points is a straight line.{{efn|name=ArchimedesStraight| As a historical note, this is an axiom of [[Archimedes]]. See e.g. Kelland (1843).<ref>{{cite book |last=Kelland |first=Philip |author-link=Philip Kelland| title=Lectures on the principles of demonstrative mathematics |year=1843 |page=58 |url=https://books.google.com/books?id=yQCFAAAAIAAJ&pg=PA58 |via=Google Books}}</ref>}}