Calculus of variations: Difference between revisions

In the 20th century [[David Hilbert]], [[Emmy Noether]], [[Leonida Tonelli]], [[Henri Lebesgue]] and [[Jacques Hadamard]] among others made significant contributions.<ref name="brunt" /> [[Marston Morse]] applied calculus of variations in what is now called [[Morse theory]].<ref name="ferguson">{{cite arXiv |last=Ferguson |first=James |eprint=math/0402357 |title= Brief Survey of the History of the Calculus of Variations and its Applications |year=2004 }}</ref> [[Lev Pontryagin]], [[R. Tyrrell Rockafellar|Ralph Rockafellar]] and F. H. Clarke developed new mathematical tools for the calculus of variations in [[optimal control theory]].<ref name="ferguson" /> The [[dynamic programming]] of [[Richard Bellman]] is an alternative to the calculus of variations.<ref>[[Dimitri Bertsekas]]. Dynamic programming and optimal control. Athena Scientific, 2005.

</ref><ref>{{cite web |title=Richard E. Bellman Control Heritage Award |year=2004 |url=http://a2c2.org/awards/richard-e-bellman-control-heritage-award |work=American Automatic Control Council |access-date=2013-07-28}}</ref>{{efn|See '''[[Harold J. Kushner]] (2004)''': regarding Dynamic Programming, "The calculus of variations had related ideas (e.g., the work of Caratheodory, the Hamilton-Jacobi equation). This led to conflicts with the calculus of variations community."}}

The calculus of variations is concerned with the maxima or minima (collectively called '''extrema''') of functionals. A functional maps [[Function (mathematics)|functions]] to [[scalar (mathematics)|scalars]], so functionals have been described as "functions of functions." Functionals have extrema with respect to the elements <math>y</math> of a given [[function space]] defined over a given [[Domain of a function|domain]]. A functional <math>J[y]</math> is said to have an extremum at the function <math>f</math>if<math>\Delta J = J[y] - J[f]</math> has the same [[Sign (mathematics)|sign]] for all <math>y</math> in an arbitrarily small neighborhood of <math>f.</math>{{efn|The neighborhood of <math>f</math> is the part of the given function space where <math>|y - f| <h</math> over the whole domain of the functions, with <math>h</math> a positive number that specifies the size of the neighborhood.<ref name='CourHilb1953P169'>{{cite book |last1=Courant |first1=R |author-link1=Richard Courant |last2=Hilbert |first2=D |author-link2=David Hilbert |title = Methods of Mathematical Physics |volume=I |edition=First English |publisher=Interscience Publishers, Inc. |year=1953 |location=New York |page=169 |isbn=978-0471504474}}</ref>}} The function <math>f</math> is called an '''extremal''' function or extremal.{{efn|name=ExtremalVsExtremum| Note the difference between the terms extremal and extremum. An extremal is a function that makes a functional an extremum.}} The extremum <math>J[f]</math> is called a local maximum if <math>\Delta J \leq 0</math> everywhere in an arbitrarily small neighborhood of <math>f,</math> and a local minimum if <math>\Delta J \geq 0</math> there. For a function space of continuous functions, extrema of corresponding functionals are called '''weak extrema''' or '''strong extrema''', depending on whether the first derivatives of the continuous functions are respectively all continuous or not.<ref name='GelfandFominPP12to13'>{{harvnb|Gelfand|Fomin|2000|pp=12–13}}</ref>

:<math>J[y] = \int_{x_1}^{x_2} L\left(x,y(x),y^{\prime}(x)\right)\, dx \, .</math>

:<math>x_1, x_2</math> are [[Constant (mathematics)|constants]],

:<math>y(x)</math> is twice continuously differentiable,

:<math>y^{\prime}(x) = \frac{dy}{dx},</math>

:<math>L\left(x, y(x), y^{\prime}(x)\right)</math> is twice continuously differentiable with respect to its arguments <math>x, y,</math> and <math>y^{\prime}.</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,

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>\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^{\prime}(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>\Phi^{\prime}(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^{\prime}\right],</math> where <math>y = f + \varepsilon \eta</math> and <math>y^{\prime} = f^{\prime} + \varepsilon \eta^{\prime}</math> are considered as functions of <math>\varepsilon</math> rather than <math>x,</math> yields

\frac{dL}{d\varepsilon}=\frac{\partial L}{\partial y}\frac{dy}{d\varepsilon} + \frac{\partial L}{\partial y^{\prime}}\frac{dy^{\prime}}{d\varepsilon}

and because &nbsp;<math>\frac{dy}{d \varepsilon} = \eta</math> &nbsp;and&nbsp; <math>\frac{d y^{\prime}}{d \varepsilon} = \eta^{\prime},</math>

\frac{dL}{d\varepsilon}=\frac{\partial L}{\partial y}\eta + \frac{\partial L}{\partial y^{\prime}}\eta^{\prime}.

& = \int_{x_1}^{x_2} \left(\frac{\partial L}{\partial f} \eta + \frac{\partial L}{\partial f^{\prime}} \eta^{\prime}\right)\, dx \\

& = \int_{x_1}^{x_2} \frac{\partial L}{\partial f} \eta \, dx + \left.\frac{\partial L}{\partial f^{\prime}} \eta \right|_{x_1}^{x_2} - \int_{x_1}^{x_2} \eta \frac{d}{dx}\frac{\partial L}{\partial f^{\prime}} \, dx \\

& = \int_{x_1}^{x_2} \left(\frac{\partial L}{\partial f} \eta - \eta \frac{d}{dx}\frac{\partial L}{\partial f^{\prime}} \right)\, dx\\

\end{align}

where <math>L\left[x, y, y^{\prime}\right] \toL\left[x, f, f^{\prime}\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>\int_{x_1}^{x_2} \eta (x) \left(\frac{\partial L}{\partial f} - \frac{d}{dx}\frac{\partial L}{\partial f^{\prime}} \right) \, dx = 0 \, .</math>

:<math>\frac{\partial L}{\partial f} -\frac{d}{dx} \frac{\partial L}{\partial f^{\prime}}=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/\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]].

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>A[y] = \int_{x_1}^{x_2} \sqrt{1 + [y^{\prime}(x) ]^2} \, dx \, ,</math>

:<math>y\,^{\prime}(x) = \frac{dy}{dx} \, , \ \ y_1=f(x_1) \, , \ \ y_2=f(x_2) \, .</math>

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>\frac{\partial L}{\partial f} -\frac{d}{dx} \frac{\partial L}{\partial f^{\prime}}=0</math>

:<math>L = \sqrt{1 + [f^{\prime}(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>\frac{d}{dx} \frac{\partial L}{\partial f^{\prime}} = 0 \, .</math>

Substituting for <math>L</math> and taking the derivative,

:<math>\frac{d}{dx} \ \frac{f^{\prime}(x)} {\sqrt{1 + [f^{\prime}(x)]^2}} \ = 0 \, .</math>

:<math>\frac{f^{\prime}(x)}{\sqrt{1+[f^{\prime}(x)]^2}} = c \, ,</math>

for some constant <math>c.</math> Then

:<math>\frac{[f^{\prime}(x)]^2}{1+[f^{\prime}(x)]^2} = c^2 \, ,</math>

:<math>[f^{\prime}(x)]^2=\frac{c^2}{1-c^2}\,</math>

:<math>f^{\prime}(x)=m</math>

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>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 = f(x).</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>}}

In physics problems it may be the case that <math>\frac{\partial L}{\partial x} = 0,</math> meaning the integrand is a function of <math>f(x)</math> and <math>f^{\prime}(x)</math> but <math>x</math> does not appear separately. In that case, the Euler–Lagrange equation can be simplified to the [[Beltrami identity]]<ref>{{cite web |author=Weisstein, Eric W. |url=http://mathworld.wolfram.com/Euler-LagrangeDifferentialEquation.html |title=Euler–Lagrange Differential Equation |website=mathworld.wolfram.com |publisher=Wolfram |at=Eq.&nbsp;(5)}}</ref>

:<math>L-f^{\prime}\frac{\partial L}{\partial f^{\prime}}=C \, ,</math>

where <math>C</math> is a constant. The left hand side is the [[Legendre transformation]] of <math>L</math> with respect to <math>f^{\prime}(x).</math>

The intuition behind this result is that, if the variable <math>x</math> is actually time, then the statement <math>\frac{\partial L}{\partial x} = 0</math> implies that the Lagrangian is time-independent. By [[Noether's theorem]], there is an associated conserved quantity. In this case, this quantity is the Hamiltonian, the Legendre transform of the Lagrangian, which (often) coincides with the energy of the system. This is (minus) the constant in Beltrami's identity.

If <math>S</math> depends on higher-derivatives of <math>y(x),</math> that is, if<blockquote><math>S = \int\limits_{a}^{b} f(x, y(x), y^{\prime}(x), ..., y^{n}(x)) dx,</math></blockquote>then <math>y</math> must satisfy the Euler–[[Siméon Denis Poisson|Poisson]] equation,<blockquote><math>\frac{\partial f}{\partial y} - \frac{d}{dx} \left( \frac{\partial f}{\partial y^{\prime}} \right) + ... + (-1)^{n} \frac{d^n}{dx^n} \left[ \frac{\partial f}{\partial y^{(n)}} \right]= 0.</math><ref>{{Cite book|last=Kot|first=Mark|title=A First Course in the Calculus of Variations|publisher=American Mathematical Society|year=2014|isbn=978-1-4704-1495-5|chapter=Chapter 4: Basic Generalizations}}</ref></blockquote>

The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integral <math>J</math> requires only first derivatives of trial functions. The condition that the first variation vanishes at an extremal may be regarded as a '''weak form''' of the Euler–Lagrange equation. The theorem of Du Bois-Reymond asserts that this weak form implies the strong form. If <math>L</math> has continuous first and second derivatives with respect to all of its arguments, and if

:<math>\frac{\partial^2 L}{\partial f'^2} \ne 0,</math>

Clearly, <math>x(t) = t^{\frac{1}{3}}</math>minimizes the functional, but we find any function <math>x \in W^{1, \infty}</math> gives a value bounded away from the infimum.

Examples (in one-dimension) are traditionally manifested across <math>W^{1,1}</math> and <math>W^{1,\infty},</math> but Ball and Mizel<ref>{{Cite journal|last=Ball & Mizel|date=1985|title=One-dimensional Variational problems whose Minimizers do not satisfy the Euler-Lagrange equation.|journal=Archive for Rational Mechanics and Analysis|volume=90|issue=4|pages=325–388|doi=10.1007/BF00276295|bibcode=1985ArRMA..90..325B|s2cid=55005550}}</ref> procured the first functional that displayed Lavrentiev's Phenomenon across <math>W^{1,p}</math> and <math>W^{1,q}</math> for <math>1 \leq p < q < \infty.</math> There are several results that gives criteria under which the phenomenon does not occur - for instance 'standard growth', a Lagrangian with no dependence on the second variable, or an approximating sequence satisfying Cesari's Condition (D) - but results are often particular, and applicable to a small class of functionals.

For example, if <math>\varphi(x, y)</math> denotes the displacement of a membrane above the domain <math>D</math> in the <math>x,y</math> plane, then its potential energy is proportional to its surface area:

:<math>U[\varphi] = \iint_D \sqrt{1 +\nabla \varphi \cdot \nabla \varphi} \,dx\,dy.\,</math>

[[Plateau's problem]] consists of finding a function that minimizes the surface area while assuming prescribed values on the boundary of <math>D</math>; the solutions are called '''minimal surfaces'''. The Euler–Lagrange equation for this problem is nonlinear:

:<math>\varphi_{xx}(1 + \varphi_y^2) + \varphi_{yy}(1 + \varphi_x^2) - 2\varphi_x \varphi_y \varphi_{xy} = 0.\,</math>

:<math>V[\varphi] = \frac{1}{2}\iint_D \nabla \varphi \cdot \nabla \varphi \, dx\, dy.\,</math>

The functional <math>V</math> is to be minimized among all trial functions <math>\varphi</math> that assume prescribed values on the boundary of <math>D.</math>If<math>u</math> is the minimizing function and <math>v</math> is an arbitrary smooth function that vanishes on the boundary of <math>D,</math> then the first variation of <math>V[u + \varepsilon v]</math> must vanish:

:<math>\frac{d}{d\varepsilon} V[u + \varepsilon v]|_{\varepsilon=0} = \iint_D \nabla u \cdot \nabla v \, dx\,dy = 0.\,</math>

:<math>\iint_D \nabla \cdot (v \nabla u) \,dx\,dy =

where <math>C</math> is the boundary of <math>D,</math> <math>s</math> is arclength along <math>C</math> and <math>\partial u / \partial n</math> is the normal derivative of <math>u</math>on<math>C.</math> Since <math>v</math> vanishes on <math>C</math> and the first variation vanishes, the result is

:<math>\iint_D v\nabla \cdot \nabla u \,dx\,dy =0 \,</math>

for all smooth functions v that vanish on the boundary of <math>D.</math> The proof for the case of one dimensional integrals may be adapted to this case to show that

:<math>\nabla \cdot \nabla u= 0 \,</math>in<math>D.</math>

:<math>W[\varphi] = \int_{-1}^{1} (x\varphi^{\prime})^2 \, dx\,</math>

among all functions <math>\varphi</math> that satisfy <math>\varphi(-1)=-1</math> and

<math>W</math> can be made arbitrarily small by choosing piecewise linear functions that make a transition between −1 and 1 in a small neighborhood of the origin. However, there is no function that makes <math>W=0.</math>{{efn|The resulting controversy over the validity of Dirichlet's principle is explained by Turnbull.<ref>{{cite web |url=http://turnbull.mcs.st-and.ac.uk/~history/Biographies/Riemann.html |title=Riemann biography |publisher=U. St. Andrew |place=UK |author=Turnbull}}</ref>}} Eventually it was shown that Dirichlet's principle is valid, but it requires a sophisticated application of the regularity theory for [[elliptic partial differential equation]]s; see Jost and Li–Jost (1998).

:<math>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.</math>

This corresponds to an external force density <math>f(x,y)</math>in<math>D,</math> an external force <math>g(s)</math> on the boundary <math>C,</math> and elastic forces with modulus <math>\sigma(s)</math>acting on <math>C.</math> The function that minimizes the potential energy '''with no restriction on its boundary values''' will be denoted by <math>u.</math> Provided that <math>f</math> and <math>g</math> are continuous, regularity theory implies that the minimizing function <math>u</math> will have two derivatives. In taking the first variation, no boundary condition need be imposed on the increment <math>v.</math>The first variation of

:<math>\iint_D \left[ \nabla u \cdot \nabla v + f v \right] \, dx\, dy + \int_C \left[ \sigma u v + g v \right] \, ds =0. \,</math>

:<math>\iint_D \left[ -v \nabla \cdot \nabla u + v f \right] \, dx \, dy + \int_C v \left[ \frac{\partial u}{\partial n} + \sigma u + g \right] \, ds =0. \,</math>

If we first set <math>v = 0</math>on<math>C,</math> the boundary integral vanishes, and we conclude as before that

:<math>- \nabla \cdot \nabla u + f =0 \,</math>

in<math>D.</math> Then if we allow <math>v</math> to assume arbitrary boundary values, this implies that <math>u</math> must satisfy the boundary condition

:<math>\frac{\partial u}{\partial n} + \sigma u + g =0, \,</math>

on<math>C.</math> This boundary condition is a consequence of the minimizing property of <math>u</math>: it is not imposed beforehand. Such conditions are called '''natural boundary conditions'''.

The preceding reasoning is not valid if <math>\sigma</math> vanishes identically on <math>C.</math> In such a case, we could allow a trial function <math>\varphi \equiv c,</math> where <math>c</math> is a constant. For such a trial function,

:<math>V[c] = c\left[ \iint_D f \, dx\,dy + \int_C g \, ds \right].</math>

By appropriate choice of <math>c,</math> <math>V</math> can assume any value unless the quantity inside the brackets vanishes. Therefore, the variational problem is meaningless unless

:<math>\iint_D f \, dx\,dy + \int_C g \, ds =0.\,</math>

:<math>Q[\varphi] = \int_{x_1}^{x_2} \left[ p(x) \varphi^{\prime}(x)^2 + q(x) \varphi(x)^2 \right] \, dx, \,</math>

where <math>\varphi</math>is restricted to functions that satisfy the boundary conditions

Let <math>R</math> be a normalization integral

:<math>R[\varphi] =\int_{x_1}^{x_2} r(x)\varphi(x)^2 \, dx.\,</math>

The functions <math>p(x)</math> and <math>r(x)</math> are required to be everywhere positive and bounded away from zero. The primary variational problem is to minimize the ratio <math>Q/R</math> among all <math>\varphi</math> satisfying the endpoint conditions. It is shown below that the Euler–Lagrange equation for the minimizing <math>u</math>is

:<math>-(pu^{\prime})^{\prime} +q u -\lambda r u = 0, \,</math>

where <math>\lambda</math> is the quotient

:<math>\lambda = \frac{Q[u]}{R[u]}. \,</math>

It can be shown (see Gelfand and Fomin 1963) that the minimizing <math>u</math> has two derivatives and satisfies the Euler–Lagrange equation. The associated <math>\lambda</math> will be denoted by <math>\lambda_1</math>; it is the lowest eigenvalue for this equation and boundary conditions. The associated minimizing function will be denoted by <math>u_1(x).</math> This variational characterization of eigenvalues leads to the [[Rayleigh–Ritz method]]: choose an approximating <math>u</math> as a linear combination of basis functions (for example trigonometric functions) and carry out a finite-dimensional minimization among such linear combinations. This method is often surprisingly accurate.

The next smallest eigenvalue and eigenfunction can be obtained by minimizing <math>Q</math> under the additional constraint

:<math>\int_{x_1}^{x_2} r(x) u_1(x) \varphi(x) \, dx=0. \,</math>

The variational problem also applies to more general boundary conditions. Instead of requiring that <math>\varphi</math> vanish at the endpoints, we may not impose any condition at the endpoints, and set

:<math>Q[\varphi] = \int_{x_1}^{x_2} \left[ p(x) \varphi^{\prime}(x)^2 + q(x)\varphi(x)^2 \right] \, dx + a_1 \varphi(x_1)^2 + a_2 \varphi(x_2)^2, \,</math>

where <math>a_1</math> and <math>a_2</math> are arbitrary. If we set <math>\varphi = u + \varepsilon v</math>the first variation for the ratio <math>Q/R</math>is

:<math>V_1 = \frac{2}{R[u]} \left( \int_{x_1}^{x_2} \left[ p(x) u^{\prime}(x)v^{\prime}(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), \,</math>

where λ is given by the ratio <math>Q[u]/R[u]</math> as previously.

:<math>\frac{R[u]}{2} V_1 = \int_{x_1}^{x_2} v(x) \left[ -(p u^{\prime})^{\prime} + q u -\lambda r u \right] \, dx + v(x_1)[ -p(x_1)u^{\prime}(x_1) + a_1 u(x_1)] + v(x_2) [p(x_2) u^{\prime}(x_2) + a_2 u(x_2)]. \,</math>

If we first require that <math>v</math> vanish at the endpoints, the first variation will vanish for all such <math>v</math> only if

:<math>-(p u^{\prime})^{\prime} + q u -\lambda r u =0 \quad \hbox{for} \quad x_1 < x < x_2.\,</math>

If<math>u</math> satisfies this condition, then the first variation will vanish for arbitrary <math>v</math> only if

:<math>-p(x_1)u^{\prime}(x_1) + a_1 u(x_1)=0, \quad \hbox{and} \quad p(x_2) u^{\prime}(x_2) + a_2 u(x_2)=0.\,</math>

Eigenvalue problems in higher dimensions are defined in analogy with the one-dimensional case. For example, given a domain <math>D</math> with boundary <math>B</math> in three dimensions we may define

:<math>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, \,</math>

:<math>R[\varphi] = \iiint_D r(X) \varphi(X)^2 \, dx \, dy \, dz.\,</math>

Let <math>u</math> be the function that minimizes the quotient <math>Q[\varphi] / R[\varphi],</math>

with no condition prescribed on the boundary <math>B.</math> The Euler–Lagrange equation satisfied by <math>u</math>is

:<math>-\nabla \cdot (p(X) \nabla u) + q(x) u - \lambda r(x) u=0,\,</math>

:<math>\lambda = \frac{Q[u]}{R[u]}.\,</math>

The minimizing <math>u</math> must also satisfy the natural boundary condition

:<math>p(S) \frac{\partial u}{\partial n} + \sigma(S) u = 0,</math>

on the boundary <math>B.</math> This result depends upon the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998) for details. Many extensions, including completeness results, asymptotic properties of the eigenvalues and results concerning the nodes of the eigenfunctions are in Courant and Hilbert (1953).

[[Fermat's principle]] states that light takes a path that (locally) minimizes the optical length between its endpoints. If the <math>x</math>-coordinate is chosen as the parameter along the path, and <math>y=f(x)</math> along the path, then the optical length is given by

:<math>A[f] = \int_{x=x_0}^{x_1} n(x,f(x)) \sqrt{1 + f^{\prime}(x)^2} dx, \,</math>

If we try <math>f(x) = f_0 (x) + \varepsilon f_1 (x)</math>

then the [[first variation]] of <math>A</math> (the derivative of <math>A</math> with respect to ε) is

:<math>\delta A[f_0,f_1] = \int_{x=x_0}^{x_1} \left[ \frac{ n(x,f_0) f_0^{\prime}(x) f_1^{\prime}(x)}{\sqrt{1 + f_0^{\prime}(x)^2}} + n_y (x,f_0) f_1 \sqrt{1 + f_0^{\prime}(x)^2} \right] dx.</math>

:<math>-\frac{d}{dx} \left[\frac{ n(x,f_0) f_0^{\prime}}{\sqrt{1 + f_0'^2}} \right] + n_y (x,f_0) \sqrt{1 + f_0^{\prime}(x)^2} =0. \,</math>

:<math>n(x,y) = n_{(-)} \quad \hbox{if} \quad x<0, \,</math>

:<math>n(x,y) = n_{(+)} \quad \hbox{if} \quad x>0,\,</math>

where <math>n_{(-)}</math> and <math>n_{(+)}</math> are constants. Then the Euler–Lagrange equation holds as before in the region where <math>x < 0</math>or<math>x > 0,</math> and in fact the path is a straight line there, since the refractive index is constant. At the <math>x = 0,</math> <math>f</math> must be continuous, but <math>f^{\prime}</math> may be discontinuous. After integration by parts in the separate regions and using the Euler–Lagrange equations, the first variation takes the form

:<math>\delta A[f_0,f_1] = f_1(0)\left[ n_{(-)}\frac{f_0^{\prime}(0^-)}{\sqrt{1 + f_0^{\prime}(0^-)^2}} - n_{(+)}\frac{f_0'(0^+)}{\sqrt{1 + f_0^{\prime}(0^+)^2}} \right].\,</math>

The factor multiplying <math>n_{(-)}</math> is the sine of angle of the incident ray with the <math>x</math> axis, and the factor multiplying <math>n_{(+)}</math> is the sine of angle of the refracted ray with the <math>x</math> axis. [[Snell's law]] for refraction requires that these terms be equal. As this calculation demonstrates, Snell's law is equivalent to vanishing of the first variation of the optical path length.

It is expedient to use vector notation: let <math>X=(x_1,x_2,x_3),</math> let <math>t</math> be a parameter, let <math>X(t)</math> be the parametric representation of a curve <math>C,</math> and let <math>\dot X(t)</math> be its tangent vector. The optical length of the curve is given by

:<math>A[C] = \int_{t=t_0}^{t_1} n(X) \sqrt{ \dot X \cdot \dot X} \, dt. \,</math>

Note that this integral is invariant with respect to changes in the parametric representation of <math>C.</math> The Euler–Lagrange equations for a minimizing curve have the symmetric form

:<math>\frac{d}{dt} P = \sqrt{ \dot X \cdot \dot X} \, \nabla n, \,</math>

:<math>P = \frac{n(X) \dot X}{\sqrt{\dot X \cdot \dot X} }.\,</math>

It follows from the definition that <math>P</math> satisfies

:<math>P \cdot P = n(X)^2. \,</math>

:<math>A[C] = \int_{t=t_0}^{t_1} P \cdot \dot X \, dt.\,</math>

This form suggests that if we can find a function <math\psi</math> whose gradient is given by <math>P,</math> then the integral <math>A</math> is given by the difference of <math\psi</math> at the endpoints of the interval of integration. Thus the problem of studying the curves that make the integral stationary can be related to the study of the level surfaces of <math\psi.</math> In order to find such a function, we turn to the wave equation, which governs the propagation of light. This formalism is used in the context of [[Lagrangian optics]] and [[Hamiltonian optics]].

:<math>u_{tt} = c^2 \nabla \cdot \nabla u, \,</math>

where <math>c</math> is the velocity, which generally depends upon <math>X.</math> Wave fronts for light are characteristic surfaces for this partial differential equation: they satisfy

:<math>\varphi_t^2 = c(X)^2 \, \nabla \varphi \cdot \nabla \varphi. \,</math>

:<math>\varphi(t,X) = t - \psi(X). \,</math>

In that case, <math\psi</math> satisfies

:<math>\nabla \psi \cdot \nabla \psi = n^2, \,</math>

where <math>n=1/c.</math> According to the theory of [[first-order partial differential equation]]s, if <math>P = \nabla \psi,</math> then <math>P</math> satisfies

:<math>\frac{dP}{ds} = n \, \nabla n,</math>

:<math>\frac{dX}{ds} = P. \,</math>

:<math>\frac{ds}{dt} = \frac{\sqrt{ \dot X \cdot \dot X} }{n}. \,</math>

We conclude that the function <math\psi</math> is the value of the minimizing integral <math>A</math> as a function of the upper end point. That is, when a family of minimizing curves is constructed, the values of the optical length satisfy the characteristic equation corresponding the wave equation. Hence, solving the associated partial differential equation of first order is equivalent to finding families of solutions of the variational problem. This is the essential content of the [[Hamilton–Jacobi theory]], which applies to more general variational problems.

In classical mechanics, the action, <math>S,</math> is defined as the time integral of the Lagrangian, <math>L.</math> The Lagrangian is the difference of energies,

:<math>L = T - U, \,</math>

where <math>T</math> is the [[kinetic energy]] of a mechanical system and <math>U</math> its [[potential energy]]. [[Hamilton's principle]] (or the action principle) states that the motion of a conservative holonomic (integrable constraints) mechanical system is such that the action integral

:<math>S = \int_{t=t_0}^{t_1} L(x, \dot x, t) \, dt</math>

is stationary with respect to variations in the path <math>x(t).</math>

:<math>\frac{d}{dt} \frac{\partial L}{\partial \dot x} = \frac{\partial L}{\partial x}, \,</math>

The conjugate momenta <math>P</math> are defined by

:<math>p = \frac{\partial L}{\partial \dot x}. \,</math>

:<math>T = \frac{1}{2} m \dot x^2, \,</math>

:<math>p = m \dot x. \,</math>

[[Hamiltonian mechanics]] results if the conjugate momenta are introduced in place of <math>\dot x</math> by a Legendre transformation of the Lagrangian <math>L</math> into the Hamiltonian <math>H</math> defined by

:<math>H(x, p, t) = p \,\dot x - L(x,\dot x, t).\,</math>

The Hamiltonian is the total energy of the system: <math>H = T + U.</math>

Analogy with Fermat's principle suggests that solutions of Lagrange's equations (the particle trajectories) may be described in terms of level surfaces of some function of <math>X.</math> This function is a solution of the [[Hamilton–Jacobi equation]]:

:<math>\frac{\partial \psi}{\partial t} + H\left(x,\frac{\partial \psi}{\partial x},t\right) = 0.\,</math>

For example, if <math>J[y]</math> is a functional with the function <math>y = y(x)</math> as its argument, and there is a small change in its argument from <math>y</math>to<math>y + h,</math> where <math>h = h(x)</math> is a function in the same function space as <math>y,</math> then the corresponding change in the functional is

:<math>\Delta J[h] = J[y+h] - J[y].</math>&nbsp;&nbsp;{{efn|name=SimplifyNotation|Note that <math>\Delta J[h]</math> and the variations below, depend on both <math>y</math> and <math>h.</math> The argument <math>y</math> has been left out to simplify the notation. For example, <math>\Delta J[h]</math> could have been written <math>\Delta J[y; h].</math><ref name='GelfandFominP12FN6'>{{harvnb | Gelfand|Fomin|2000 | p=12, footnote 6}}</ref>}}

The functional <math>J[y]</math> is said to be '''differentiable''' if

:<math>\Delta J[h] = \varphi [h] + \varepsilon \|h\|,</math>

where <math>\varphi[h]</math> is a linear functional,{{efn|name=Linear|A functional <math>\varphi[h]</math> is said to be '''linear''' if <math>\varphi[\alpha h] = \alpha \varphi[h]</math> &nbsp; and &nbsp; <math>\varphi\left[h + h_2\right] = \varphi[h] + \varphi\left[h_2\right],</math> where <math>h, h_2</math> are functions and <math>\alpha</math> is a real number.<ref name='GelfandFominP8'>{{harvnb | Gelfand|Fomin| 2000 | p=8 }}</ref>}} <math>\|h\|</math> is the norm of <math>h,</math>{{efn|name=Norm| For a function <math>h = h(x)</math> that is defined for <math>a\leqx\leq b,</math> where <math>a</math> and <math>b</math> are real numbers, the norm of <math>h</math> is its maximum absolute value, i.e. <math>\|h\| = \displaystyle\max_{a \leqx\leq b} |h(x)|.</math><ref name='GelfandFominP6'>{{harvnb | Gelfand|Fomin| 2000 | p=6 }}</ref>}} and <math>\varepsilon \to 0</math>as<math>\|h\| \to 0.</math> The linear functional <math>\varphi[h]</math> is the first variation of <math>J[y]</math> and is denoted by,<ref name='GelfandFominP11–12'>{{harvnb | Gelfand|Fomin| 2000 | pp=11–12}}</ref>