Calculus of variations explained

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functionsand functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.

A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends upon the material of the medium. One corresponding concept in mechanics is the principle of least/stationary action.

Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet's principle. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in soapy water. Although such experiments are relatively easy to perform, their mathematical formulation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.

History

The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696).^[1] It immediately occupied the attention of Jacob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733. Lagrange was influenced by Euler's work to contribute significantly to the theory. After Euler saw the 1755 work of the 19-year-old Lagrange, Euler dropped his own partly geometric approach in favor of Lagrange's purely analytic approach and renamed the subject the calculus of variations in his 1756 lecture Elementa Calculi Variationum.^{[2] [3]}

Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. Isaac Newton and Gottfried Leibniz also gave some early attention to the subject.^[4] To this discrimination Vincenzo Brunacci (1810), Carl Friedrich Gauss (1829), Siméon Poisson (1831), Mikhail Ostrogradsky (1834), and Carl Jacobi (1837) have been among the contributors. An important general work is that of Sarrus (1842) which was condensed and improved by Cauchy (1844). Other valuable treatises and memoirs have been written by Strauch (1849), Jellett (1850), Otto Hesse (1857), Alfred Clebsch (1858), and Lewis Buffett Carll (1885), but perhaps the most important work of the century is that of Weierstrass. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation. The 20th and the 23rd Hilbert problem published in 1900 encouraged further development.

In the 20th century David Hilbert, Oskar Bolza, Gilbert Ames Bliss, Emmy Noether, Leonida Tonelli, Henri Lebesgue and Jacques Hadamard among others made significant contributions. Marston Morse applied calculus of variations in what is now called Morse theory.^[5] Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory. The dynamic programming of Richard Bellman is an alternative to the calculus of variations.^{[6] [7] [8]}

Extrema

The calculus of variations is concerned with the maxima or minima (collectively called extrema) of functionals. A functional maps functions to scalars, so functionals have been described as "functions of functions." Functionals have extrema with respect to the elements

y of a given function space defined over a given domain. A functional J[y] is said to have an extremum at the function f if \Delta J = J[y] - J[f] has the same sign for all y in an arbitrarily small neighborhood of f. The function f is called an extremal function or extremal. The extremum J[f] is called a local maximum if \Delta J \leq 0 everywhere in an arbitrarily small neighborhood of f, and a local minimum if \Delta J \geq 0 there. For a function space of continuous functions, extrema of corresponding functionals are called strong extrema or weak extrema, depending on whether the first derivatives of the continuous functions are respectively all continuous or not.

Both strong and weak extrema of functionals are for a space of continuous functions but strong extrema have the additional requirement that the first derivatives of the functions in the space be continuous. Thus a strong extremum is also a weak extremum, but the converse may not hold. Finding strong extrema is more difficult than finding weak extrema. An example of a necessary condition that is used for finding weak extrema is the Euler–Lagrange equation.

Euler–Lagrange equation

See main article: 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.

Consider the functional$$J[y]\; =\; \backslash int\_^\; L\backslash left(x,y(x),y\text{'}(x)\backslash right)\backslash ,\; dx\; \backslash ,\; .$$where

x _{1,} x _{2} are constants,y(x) is twice continuously differentiable,y'(x) = (dy)/(dx) , L\left(x, y(x), y'(x)\right) is twice continuously differentiable with respect to its arguments x, y, and y'.

If the functional

J[y] attains a local minimum at f, and η (x) is an arbitrary function that has at least one derivative and vanishes at the endpoints x _{1} and x _{2,} then for any number \varepsilon close to 0,$$J[f]\; \backslash le\; J[f\; +\; \backslash varepsilon\; \backslash eta]\; \backslash ,\; .$$

The term

\varepsilon η is called the variation of the function f and is denoted by \delta f.

Substituting

f + \varepsilon η for y in the functional J[y], the result is a function of \varepsilon,

$$\backslash Phi(\backslash varepsilon)\; =\; J[f+\backslash varepsilon\backslash eta]\; \backslash ,\; .$$Since the functional

J[y] has a minimum for y = f the function \Phi(\varepsilon) has a minimum at \varepsilon = 0 and thus,$$\backslash Phi\text{'}(0)\; \backslash equiv\; \backslash left.\backslash frac\backslash right|\_\; =\; \backslash int\_^\; \backslash left.\backslash frac\backslash right|\_\; dx\; =\; 0\; \backslash ,\; .$$

Taking the total derivative of

L\left[x, y, y'\right], where y = f + \varepsilon η and y' = f' + \varepsilon η ' are considered as functions of \varepsilon rather than x, yields$$\backslash frac=\backslash frac\backslash frac\; +\; \backslash frac\backslash frac$$and because (dy)/(d \varepsilon) = η and (d y')/(d \varepsilon) = η ', $$\backslash frac=\backslash frac\backslash eta\; +\; \backslash frac\backslash eta\text{'}.$$

Therefore,where

L\left[x, y, y'\right] \to L\left[x, f, f'\right] when \varepsilon = 0 and we have used integration by parts on the second term. The second term on the second line vanishes because η = 0 at x _{1} and x _{2} by definition. Also, as previously mentioned the left side of the equation is zero so that$$\backslash int\_^\; \backslash eta\; (x)\; \backslash left(\backslash frac\; -\; \backslash frac\backslash frac\; \backslash right)\; \backslash ,\; dx\; =\; 0\; \backslash ,\; .$$

According to the fundamental lemma of calculus of variations, the part of the integrand in parentheses is zero, i.e.$$\backslash frac\; -\backslash frac\; \backslash frac=0$$which is called the Euler–Lagrange equation. The left hand side of this equation is called the functional derivative of

J[f] and is denoted \delta J/\delta f(x).

In general this gives a second-order ordinary differential equation which can be solved to obtain the extremal function

f(x). The Euler–Lagrange equation is a necessary, but not sufficient, condition for an extremum J[f]. A sufficient condition for a minimum is given in the section Variations and sufficient condition for a minimum.

Example

In order to illustrate this process, consider the problem of finding the extremal function

y = f(x), which is the shortest curve that connects two points \left(x _{1,} y _{1\right)} and \left(x _{2,} y _{2\right).} The arc length of the curve is given by$$A[y]\; =\; \backslash int\_^\; \backslash sqrt\; \backslash ,\; dx\; \backslash ,,$$with$$y\text{'}(x)\; =\; \backslash frac\; \backslash ,,\; \backslash \; \backslash \; y\_1=f(x\_1)\; \backslash ,,\; \backslash \; \backslash \; y\_2=f(x\_2)\; \backslash ,\; .$$Note that assuming is a function of 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

f(x) that minimizes the functional A[y]. $$\backslash frac\; -\backslash frac\; \backslash frac=0$$with$$L\; =\; \backslash sqrt\; \backslash ,\; .$$

Since

f does not appear explicitly in L, the first term in the Euler–Lagrange equation vanishes for all f(x) and thus,$$\backslash frac\; \backslash frac\; =\; 0\; \backslash ,\; .$$Substituting for L and taking the derivative,$$\backslash frac\; \backslash \; \backslash frac\; \backslash \; =\; 0\; \backslash ,\; .$$

Thus$$\backslash frac\; =\; c\; \backslash ,,$$for some constant

c. Then$$\backslash frac\; =\; c^2\; \backslash ,,$$whereSolving, we get$$[f\text{'}(x)]^2=\backslash frac$$which implies that$$f\text{'}(x)=m$$is a constant and therefore that the shortest curve that connects two points \left(x _{1,} y _{1\right)} and \left(x _{2,} y _{2\right)} is$$f(x)\; =\; m\; x\; +\; b\; \backslash qquad\; \backslash text\; \backslash \; \backslash \; m\; =\; \backslash frac\; \backslash quad\; \backslash text\; \backslash quad\; b\; =\; \backslash frac$$and we have thus found the extremal function f(x) that minimizes the functional A[y] so that A[f] is a minimum. The equation for a straight line is y = f(x). In other words, the shortest distance between two points is a straight line.

Beltrami's identity

In physics problems it may be the case that

(\partial L)/(\partial x) = 0 , meaning the integrand is a function of f(x) and f'(x) but x does not appear separately. In that case, the Euler–Lagrange equation can be simplified to the Beltrami identity^[9] $$L\; -\; f\text{'}\; \backslash frac\; =\; C\; \backslash ,,$$where C is a constant. The left hand side is the Legendre transformation of L with respect to f'(x).

The intuition behind this result is that, if the variable

x is actually time, then the statement (\partial L)/(\partial x) = 0 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.

Euler–Poisson equation

If

S depends on higher-derivatives of y(x), that is, if $$S\; =\; \backslash int\_^\; f(x,\; y(x),\; y\text{'}(x),\; \backslash dots,\; y^(x))\; dx,$$ then y must satisfy the Euler–Poisson equation,^[10] $$\backslash frac\; -\; \backslash frac\; \backslash left(\backslash frac\; \backslash right)\; +\; \backslash dots\; +\; (-1)^\; \backslash frac\; \backslash left[\backslash frac\{\backslash partial\; f\}\{\backslash partial\; y^\{(n)\}\}\; \backslash right]=\; 0.$$

Du Bois-Reymond's theorem

The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integral

J 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 L has continuous first and second derivatives with respect to all of its arguments, and if$$\backslash frac\; \backslash ne\; 0,$$then f has two continuous derivatives, and it satisfies the Euler–Lagrange equation.

Lavrentiev phenomenon

Hilbert was the first to give good conditions for the Euler–Lagrange equations to give a stationary solution. Within a convex area and a positive thrice differentiable Lagrangian the solutions are composed of a countable collection of sections that either go along the boundary or satisfy the Euler–Lagrange equations in the interior.

However Lavrentiev in 1926 showed that there are circumstances where there is no optimum solution but one can be approached arbitrarily closely by increasing numbers of sections. The Lavrentiev Phenomenon identifies a difference in the infimum of a minimization problem across different classes of admissible functions. For instance the following problem, presented by Manià in 1934:^[11] $$L[x]\; =\; \backslash int\_0^1\; (x^3-t)^2\; x\text{'}^6,$$$$=\; \backslash .$$

Clearly,

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

Examples (in one-dimension) are traditionally manifested across

W ^{1,1} and W ^{1,infty} , but Ball and Mizel^[12] procured the first functional that displayed Lavrentiev's Phenomenon across W ^{1,p} and W ^{1,q} for 1 \leq p < q < inf ty. 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.

Connected with the Lavrentiev Phenomenon is the repulsion property: any functional displaying Lavrentiev's Phenomenon will display the weak repulsion property.^[13]

Functions of several variables

For example, if

\varphi(x, y) denotes the displacement of a membrane above the domain D in the x,y plane, then its potential energy is proportional to its surface area:$$U[\backslash varphi]\; =\; \backslash iint\_D\; \backslash sqrt\; \backslash ,dx\backslash ,dy.$$Plateau's problem consists of finding a function that minimizes the surface area while assuming prescribed values on the boundary of D ; the solutions are called minimal surfaces. The Euler–Lagrange equation for this problem is nonlinear:$$\backslash varphi\_(1\; +\; \backslash varphi\_y^2)\; +\; \backslash varphi\_(1\; +\; \backslash varphi\_x^2)\; -\; 2\backslash varphi\_x\; \backslash varphi\_y\; \backslash varphi\_\; =\; 0.$$See Courant (1950) for details.

Dirichlet's principle

It is often sufficient to consider only small displacements of the membrane, whose energy difference from no displacement is approximated by$$V[\backslash varphi]\; =\; \backslash frac\backslash iint\_D\; \backslash nabla\; \backslash varphi\; \backslash cdot\; \backslash nabla\; \backslash varphi\; \backslash ,\; dx\backslash ,\; dy.$$The functional

V is to be minimized among all trial functions \varphi that assume prescribed values on the boundary of D. If u is the minimizing function and v is an arbitrary smooth function that vanishes on the boundary of D, then the first variation of V[u + \varepsilon v] must vanish:$$\backslash left.\backslash frac\; V[u\; +\; \backslash varepsilon\; v]\backslash right|\_\; =\; \backslash iint\_D\; \backslash nabla\; u\; \backslash cdot\; \backslash nabla\; v\; \backslash ,\; dx\backslash ,dy\; =\; 0.$$Provided that u has two derivatives, we may apply the divergence theorem to obtain$$\backslash iint\_D\; \backslash nabla\; \backslash cdot\; (v\; \backslash nabla\; u)\; \backslash ,dx\backslash ,dy\; =\backslash iint\_D\; \backslash nabla\; u\; \backslash cdot\; \backslash nabla\; v\; +\; v\; \backslash nabla\; \backslash cdot\; \backslash nabla\; u\; \backslash ,dx\backslash ,dy\; =\; \backslash int\_C\; v\; \backslash frac\; \backslash ,\; ds,$$where C is the boundary of D, s is arclength along C and \partial u / \partial n is the normal derivative of u on C. Since v vanishes on C and the first variation vanishes, the result is$$\backslash iint\_D\; v\backslash nabla\; \backslash cdot\; \backslash nabla\; u\; \backslash ,dx\backslash ,dy\; =0$$for all smooth functions v that vanish on the boundary of D. The proof for the case of one dimensional integrals may be adapted to this case to show that$$\backslash nabla\; \backslash cdot\; \backslash nabla\; u=\; 0$$in D.

The difficulty with this reasoning is the assumption that the minimizing function

u must have two derivatives. Riemann argued that the existence of a smooth minimizing function was assured by the connection with the physical problem: membranes do indeed assume configurations with minimal potential energy. Riemann named this idea the Dirichlet principle in honor of his teacher Peter Gustav Lejeune Dirichlet. However Weierstrass gave an example of a variational problem with no solution: minimize$$W[\backslash varphi]\; =\; \backslash int\_^\; (x\backslash varphi\text{'})^2\; \backslash ,\; dx$$among all functions \varphi that satisfy \varphi( -1 ) =-1 and \varphi( 1 ) =1 . W 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 W =0 . Eventually it was shown that Dirichlet's principle is valid, but it requires a sophisticated application of the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998).

Generalization to other boundary value problems

A more general expression for the potential energy of a membrane is$$V[\backslash varphi]\; =\; \backslash iint\_D\; \backslash left[\backslash frac\{1\}\{2\}\; \backslash nabla\; \backslash varphi\; \backslash cdot\; \backslash nabla\; \backslash varphi\; +\; f(x,y)\; \backslash varphi\; \backslash right]\; \backslash ,\; dx\backslash ,dy\; \backslash ,\; +\; \backslash int\_C\; \backslash left[\backslash frac\{1\}\{2\}\; \backslash sigma(s)\; \backslash varphi^2\; +\; g(s)\; \backslash varphi\; \backslash right]\; \backslash ,\; ds.$$This corresponds to an external force density

f(x,y) in D, an external force g(s) on the boundary C, and elastic forces with modulus \sigma(s) acting on C. The function that minimizes the potential energy with no restriction on its boundary values will be denoted by u. Provided that f and g are continuous, regularity theory implies that the minimizing function u will have two derivatives. In taking the first variation, no boundary condition need be imposed on the increment v. The first variation of V[u + \varepsilon v] is given by$$\backslash iint\_D\; \backslash left[\backslash nabla\; u\; \backslash cdot\; \backslash nabla\; v\; +\; f\; v\; \backslash right]\; \backslash ,\; dx\backslash ,\; dy\; +\; \backslash int\_C\; \backslash left[\backslash sigma\; u\; v\; +\; g\; v\; \backslash right]\; \backslash ,\; ds\; =\; 0.$$If we apply the divergence theorem, the result is$$\backslash iint\_D\; \backslash left[-v\; \backslash nabla\; \backslash cdot\; \backslash nabla\; u\; +\; v\; f\; \backslash right]\; \backslash ,\; dx\; \backslash ,\; dy\; +\; \backslash int\_C\; v\; \backslash left[\backslash frac\{\backslash partial\; u\}\{\backslash partial\; n\}\; +\; \backslash sigma\; u\; +\; g\; \backslash right]\; \backslash ,\; ds\; =0.$$If we first set v = 0 on C, the boundary integral vanishes, and we conclude as before that$$-\; \backslash nabla\; \backslash cdot\; \backslash nabla\; u\; +\; f\; =0$$in D. Then if we allow v to assume arbitrary boundary values, this implies that u must satisfy the boundary condition$$\backslash frac\; +\; \backslash sigma\; u\; +\; g\; =0,$$on C. This boundary condition is a consequence of the minimizing property of u : it is not imposed beforehand. Such conditions are called natural boundary conditions.

The preceding reasoning is not valid if

\sigma vanishes identically on C. In such a case, we could allow a trial function \varphi \equiv c, where c is a constant. For such a trial function,$$V[c]\; =\; c\backslash left[\backslash iint\_D\; f\; \backslash ,\; dx\backslash ,dy\; +\; \backslash int\_C\; g\; \backslash ,\; ds\; \backslash right].$$By appropriate choice of c, V can assume any value unless the quantity inside the brackets vanishes. Therefore, the variational problem is meaningless unless$$\backslash iint\_D\; f\; \backslash ,\; dx\backslash ,dy\; +\; \backslash int\_C\; g\; \backslash ,\; ds\; =0.$$This condition implies that net external forces on the system are in equilibrium. If these forces are in equilibrium, then the variational problem has a solution, but it is not unique, since an arbitrary constant may be added. Further details and examples are in Courant and Hilbert (1953).

Eigenvalue problems

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

Sturm–Liouville problems

See also: Sturm–Liouville theory. The Sturm–Liouville eigenvalue problem involves a general quadratic form$$Q[y]\; =\; \backslash int\_^\; \backslash left[p(x)\; y\text{'}(x)^2\; +\; q(x)\; y(x)^2\; \backslash right]\; \backslash ,\; dx,$$where

y is restricted to functions that satisfy the boundary conditions$$y(x\_1)=0,\; \backslash quad\; y(x\_2)=0.$$Let R be a normalization integral$$R[y]\; =\backslash int\_^\; r(x)y(x)^2\; \backslash ,\; dx.$$The functions p(x) and r(x) are required to be everywhere positive and bounded away from zero. The primary variational problem is to minimize the ratio Q/R among all y satisfying the endpoint conditions, which is equivalent to minimizing Q[y] under the constraint that R[y] is constant. It is shown below that the Euler–Lagrange equation for the minimizing u is$$-(p\; u\text{'})\text{'}\; +q\; u\; -\backslash lambda\; r\; u\; =\; 0,$$where λ is the quotient$$\backslash lambda\; =\; \backslash frac.$$It can be shown (see Gelfand and Fomin 1963) that the minimizing u has two derivatives and satisfies the Euler–Lagrange equation. The associated λ will be denoted by λ _{1} ; it is the lowest eigenvalue for this equation and boundary conditions. The associated minimizing function will be denoted by u _{1(x).} This variational characterization of eigenvalues leads to the Rayleigh–Ritz method: choose an approximating u 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

Q under the additional constraint$$\backslash int\_^\; r(x)\; u\_1(x)\; y(x)\; \backslash ,\; dx\; =\; 0.$$This procedure can be extended to obtain the complete sequence of eigenvalues and eigenfunctions for the problem.

The variational problem also applies to more general boundary conditions. Instead of requiring that

y vanish at the endpoints, we may not impose any condition at the endpoints, and set$$Q[y]\; =\; \backslash int\_^\; \backslash left[p(x)\; y\text{'}(x)^2\; +\; q(x)y(x)^2\; \backslash right]\; \backslash ,\; dx\; +\; a\_1\; y(x\_1)^2\; +\; a\_2\; y(x\_2)^2,$$where a _{1} and a _{2} are arbitrary. If we set y = u + \varepsilon v , the first variation for the ratio Q/R is$$V\_1\; =\; \backslash frac\; \backslash left(\backslash int\_^\; \backslash left[p(x)\; u\text{'}(x)v\text{'}(x)\; +\; q(x)u(x)v(x)\; -\backslash lambda\; r(x)\; u(x)\; v(x)\; \backslash right]\; \backslash ,\; dx\; +\; a\_1\; u(x\_1)v(x\_1)\; +\; a\_2\; u(x\_2)v(x\_2)\; \backslash right),$$where λ is given by the ratio Q[u]/R[u] as previously.After integration by parts,$$\backslash frac\; V\_1\; =\; \backslash int\_^\; v(x)\; \backslash left[-(p\; u\text{'})\text{'}\; +\; q\; u\; -\backslash lambda\; r\; u\; \backslash right]\; \backslash ,\; dx\; +\; v(x\_1)[-p(x\_1)u\text{'}(x\_1)\; +\; a\_1\; u(x\_1)]\; +\; v(x\_2)\; [p(x\_2)\; u\text{'}(x\_2)\; +\; a\_2\; u(x\_2)].$$If we first require that v vanish at the endpoints, the first variation will vanish for all such v only ifIf u satisfies this condition, then the first variation will vanish for arbitrary v only if$$-p(x\_1)u\text{'}(x\_1)\; +\; a\_1\; u(x\_1)=0,\; \backslash quad\; \backslash hbox\; \backslash quad\; p(x\_2)\; u\text{'}(x\_2)\; +\; a\_2\; u(x\_2)=0.$$These latter conditions are the natural boundary conditions for this problem, since they are not imposed on trial functions for the minimization, but are instead a consequence of the minimization.

Eigenvalue problems in several dimensions

Eigenvalue problems in higher dimensions are defined in analogy with the one-dimensional case. For example, given a domain

D with boundary B in three dimensions we may define$$Q[\backslash varphi]\; =\; \backslash iiint\_D\; p(X)\; \backslash nabla\; \backslash varphi\; \backslash cdot\; \backslash nabla\; \backslash varphi\; +\; q(X)\; \backslash varphi^2\; \backslash ,\; dx\; \backslash ,\; dy\; \backslash ,\; dz\; +\; \backslash iint\_B\; \backslash sigma(S)\; \backslash varphi^2\; \backslash ,\; dS,$$and$$R[\backslash varphi]\; =\; \backslash iiint\_D\; r(X)\; \backslash varphi(X)^2\; \backslash ,\; dx\; \backslash ,\; dy\; \backslash ,\; dz.$$Let u be the function that minimizes the quotient Q[\varphi] / R[\varphi], with no condition prescribed on the boundary B. The Euler–Lagrange equation satisfied by u is$$-\backslash nabla\; \backslash cdot\; (p(X)\; \backslash nabla\; u)\; +\; q(x)\; u\; -\; \backslash lambda\; r(x)\; u=0,$$where$$\backslash lambda\; =\; \backslash frac.$$The minimizing u must also satisfy the natural boundary condition$$p(S)\; \backslash frac\; +\; \backslash sigma(S)\; u\; =\; 0,$$on the boundary B. 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).

Applications

Optics

Fermat's principle states that light takes a path that (locally) minimizes the optical length between its endpoints. If the

x -coordinate is chosen as the parameter along the path, and y = f(x) along the path, then the optical length is given by$$A[f]\; =\; \backslash int\_^\; n(x,f(x))\; \backslash sqrt\; dx,$$where the refractive index n(x,y) depends upon the material.If we try f(x) = f _{0} (x) + \varepsilon f _{1} (x) then the first variation of A (the derivative of A with respect to ε) is$$\backslash delta\; A[f\_0,f\_1]\; =\; \backslash int\_^\; \backslash left[\backslash frac\{\; n(x,f\_0)\; f\_0\text{'}(x)\; f\_1\text{'}(x)\}\{\backslash sqrt\{1\; +\; f\_0\text{'}(x)^2\}\}\; +\; n\_y\; (x,f\_0)\; f\_1\; \backslash sqrt\{1\; +\; f\_0\text{'}(x)^2\}\; \backslash right]\; dx.$$

After integration by parts of the first term within brackets, we obtain the Euler–Lagrange equation$$-\backslash frac\; \backslash left[\backslash frac\{\; n(x,f\_0)\; f\_0\text{'}\}\{\backslash sqrt\{1\; +\; f\_0\text{'}^2\}\}\; \backslash right]\; +\; n\_y\; (x,f\_0)\; \backslash sqrt\; =\; 0.$$

The light rays may be determined by integrating this equation. This formalism is used in the context of Lagrangian optics and Hamiltonian optics.

Snell's law

There is a discontinuity of the refractive index when light enters or leaves a lens. Letwhere

n _{(-)} and n _{(+)} are constants. Then the Euler–Lagrange equation holds as before in the region where x < 0 or x > 0 , and in fact the path is a straight line there, since the refractive index is constant. At the x = 0 , f must be continuous, but f' may be discontinuous. After integration by parts in the separate regions and using the Euler–Lagrange equations, the first variation takes the form$$\backslash delta\; A[f\_0,f\_1]\; =\; f\_1(0)\backslash left[n\_\{(-)\}\backslash frac\{f\_0\text{'}(0^-)\}\{\backslash sqrt\{1\; +\; f\_0\text{'}(0^-)^2\}\}\; -\; n\_\{(+)\}\backslash frac\{f\_0\text{'}(0^+)\}\{\backslash sqrt\{1\; +\; f\_0\text{'}(0^+)^2\}\}\; \backslash right].$$

The factor multiplying

n _{(-)} is the sine of angle of the incident ray with the x axis, and the factor multiplying n _{(+)} is the sine of angle of the refracted ray with the x 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.

Fermat's principle in three dimensions

It is expedient to use vector notation: let

X = (x _{1,x} _{2,x} _{3),} let t be a parameter, let X(t) be the parametric representation of a curve C, and let • X(t) be its tangent vector. The optical length of the curve is given by$$A[C]\; =\; \backslash int\_^\; n(X)\; \backslash sqrt\; \backslash ,\; dt.$$

Note that this integral is invariant with respect to changes in the parametric representation of

C. The Euler–Lagrange equations for a minimizing curve have the symmetric form$$\backslash frac\; P\; =\; \backslash sqrt\; \backslash ,\; \backslash nabla\; n,$$where$$P\; =\; \backslash frac.$$

It follows from the definition that

P satisfies$$P\; \backslash cdot\; P\; =\; n(X)^2.$$

Therefore, the integral may also be written as$$A[C]\; =\; \backslash int\_^\; P\; \backslash cdot\; \backslash dot\; X\; \backslash ,\; dt.$$

This form suggests that if we can find a function

\psi whose gradient is given by P, then the integral A is given by the difference of \psi 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 \psi. 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.

Connection with the wave equation

The wave equation for an inhomogeneous medium is$$u\_\; =\; c^2\; \backslash nabla\; \backslash cdot\; \backslash nabla\; u,$$where

c is the velocity, which generally depends upon X. Wave fronts for light are characteristic surfaces for this partial differential equation: they satisfy$$\backslash varphi\_t^2\; =\; c(X)^2\; \backslash ,\; \backslash nabla\; \backslash varphi\; \backslash cdot\; \backslash nabla\; \backslash varphi.$$

We may look for solutions in the form$$\backslash varphi(t,X)\; =\; t\; -\; \backslash psi(X).$$

In that case,

\psi satisfies$$\backslash nabla\; \backslash psi\; \backslash cdot\; \backslash nabla\; \backslash psi\; =\; n^2,$$where n =1 /c. According to the theory of first-order partial differential equations, if P = \nabla \psi, then P satisfies$$\backslash frac\; =\; n\; \backslash ,\; \backslash nabla\; n,$$along a system of curves (the light rays) that are given by$$\backslash frac\; =\; P.$$

These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification$$\backslash frac\; =\; \backslash frac.$$

We conclude that the function

\psi is the value of the minimizing integral A 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.

Mechanics

See main article: Action (physics). In classical mechanics, the action,

S, is defined as the time integral of the Lagrangian, L. The Lagrangian is the difference of energies,$$L\; =\; T\; -\; U,$$where T is the kinetic energy of a mechanical system and U 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$$S\; =\; \backslash int\_^\; L(x,\; \backslash dot\; x,\; t)\; \backslash ,\; dt$$is stationary with respect to variations in the path x(t). The Euler–Lagrange equations for this system are known as Lagrange's equations:$$\backslash frac\; \backslash frac\; =\; \backslash frac,$$and they are equivalent to Newton's equations of motion (for such systems).

The conjugate momenta

P are defined by$$p\; =\; \backslash frac.$$For example, if$$T\; =\; \backslash frac\; m\; \backslash dot\; x^2,$$then $$p\; =\; m\; \backslash dot\; x.$$Hamiltonian mechanics results if the conjugate momenta are introduced in place of • x by a Legendre transformation of the Lagrangian L into the Hamiltonian H defined by$$H(x,\; p,\; t)\; =\; p\; \backslash ,\backslash dot\; x\; -\; L(x,\backslash dot\; x,\; t).$$The Hamiltonian is the total energy of the system: H = T + U. 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 X. This function is a solution of the Hamilton–Jacobi equation:$$\backslash frac\; +\; H\backslash left(x,\backslash frac,t\backslash right)\; =\; 0.$$

Further applications

Further applications of the calculus of variations include the following:

• The derivation of the catenary shape
• Solution to Newton's minimal resistance problem
• Solution to the brachistochrone problem
• Solution to the tautochrone problem
• Solution to isoperimetric problems
• Calculating geodesics
• Finding minimal surfaces and solving Plateau's problem
• Optimal control
• Analytical mechanics, or reformulations of Newton's laws of motion, most notably Lagrangian and Hamiltonian mechanics;
• Geometric optics, especially Lagrangian and Hamiltonian optics;
• Variational method (quantum mechanics), one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states;
• Variational Bayesian methods, a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning;
• Variational methods in general relativity, a family of techniques using calculus of variations to solve problems in Einstein's general theory of relativity;
• Finite element method is a variational method for finding numerical solutions to boundary-value problems in differential equations;
• Total variation denoising, an image processing method for filtering high variance or noisy signals.

Variations and sufficient condition for a minimum

Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. 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[y] is a functional with the function y = y(x) as its argument, and there is a small change in its argument from y to y + h, where h = h(x) is a function in the same function space as y, then the corresponding change in the functional is$$\backslash Delta\; J[h]\; =\; J[y+h]\; -\; J[y].$$

The functional

J[y] is said to be differentiable if$$\backslash Delta\; J[h]\; =\; \backslash varphi\; [h]\; +\; \backslash varepsilon\; \backslash |h\backslash |,$$where \varphi[h] is a linear functional, \|h\| is the norm of h, and \varepsilon \to 0 as \|h\| \to 0 . The linear functional \varphi[h] is the first variation of J[y] and is denoted by,$$\backslash delta\; J[h]\; =\; \backslash varphi[h].$$

The functional

J[y] is said to be twice differentiable if$$\backslash Delta\; J[h]\; =\; \backslash varphi\_1\; [h]\; +\; \backslash varphi\_2\; [h]\; +\; \backslash varepsilon\; \backslash |h\backslash |^2,$$where \varphi _{1[h]} is a linear functional (the first variation), \varphi _{2[h]} is a quadratic functional, and \varepsilon \to 0 as \|h\| \to 0 . The quadratic functional \varphi _{2[h]} is the second variation of J[y] and is denoted by,$$\backslash delta^2\; J[h]\; =\; \backslash varphi\_2[h].$$

The second variation

\delta ^{2} J[h] is said to be strongly positive if$$\backslash delta^2J[h]\; \backslash ge\; k\; \backslash |h\backslash |^2,$$for all h and for some constant k > 0 .

Using the above definitions, especially the definitions of first variation, second variation, and strongly positive, the following sufficient condition for a minimum of a functional can be stated.

See also

• First variation
• Isoperimetric inequality
• Variational principle
• Variational bicomplex
• Fermat's principle
• Principle of least action
• Infinite-dimensional optimization
• Finite element method
• Functional analysis
• Ekeland's variational principle
• Inverse problem for Lagrangian mechanics
• Obstacle problem
• Perturbation methods
• Young measure
• Optimal control
• Direct method in calculus of variations
• Noether's theorem
• De Donder–Weyl theory
• Variational Bayesian methods
• Chaplygin problem
• Nehari manifold
• Hu–Washizu principle
• Luke's variational principle
• Mountain pass theorem
  • Measures of central tendency as solutions to variational problems
• Stampacchia Medal
• Fermat Prize
• Convenient vector space

Further reading

• Benesova, B. and Kruzik, M.: "Weak Lower Semicontinuity of Integral Functionals and Applications". SIAM Review 59(4) (2017), 703–766.
• Bolza, O.: Lectures on the Calculus of Variations. Chelsea Publishing Company, 1904, available on Digital Mathematics library. 2nd edition republished in 1961, paperback in 2005, .
• Cassel, Kevin W.: Variational Methods with Applications in Science and Engineering, Cambridge University Press, 2013.
• Clegg, J.C.: Calculus of Variations, Interscience Publishers Inc., 1968.
• Courant, R.: Dirichlet's principle, conformal mapping and minimal surfaces. Interscience, 1950.
• Dacorogna, Bernard: "Introduction" Introduction to the Calculus of Variations, 3rd edition. 2014, World Scientific Publishing, .
• Elsgolc, L.E.: Calculus of Variations, Pergamon Press Ltd., 1962.
• Forsyth, A.R.: Calculus of Variations, Dover, 1960.
• Fox, Charles: An Introduction to the Calculus of Variations, Dover Publ., 1987.
• Giaquinta, Mariano; Hildebrandt, Stefan: Calculus of Variations I and II, Springer-Verlag, and
• Jost, J. and X. Li-Jost: Calculus of Variations. Cambridge University Press, 1998.
• Lebedev, L.P. and Cloud, M.J.: The Calculus of Variations and Functional Analysis with Optimal Control and Applications in Mechanics, World Scientific, 2003, pages 1–98.
• Logan, J. David: Applied Mathematics, 3rd edition. Wiley-Interscience, 2006
• Book: Pike, Ralph W. . http://www.mpri.lsu.edu/textbook/Chapter8-b.htm. Chapter 8: Calculus of Variations. Optimization for Engineering Systems. Louisiana State University. https://web.archive.org/web/20070705141725/http://www.mpri.lsu.edu/bookindex.html. dead. 2007-07-05.
• Roubicek, T.: "Calculus of variations". Chap.17 in: Mathematical Tools for Physicists. (Ed. M. Grinfeld) J. Wiley, Weinheim, 2014,, pp. 551–588.
• Sagan, Hans: Introduction to the Calculus of Variations, Dover, 1992.
• Weinstock, Robert: Calculus of Variations with Applications to Physics and Engineering, Dover, 1974 (reprint of 1952 ed.).

External links

• Variational calculus. Encyclopedia of Mathematics.
• calculus of variations. PlanetMath.
• Calculus of Variations. MathWorld.
• Calculus of variations. Example problems.
• Mathematics - Calculus of Variations and Integral Equations. Lectures on YouTube.
• Selected papers on Geodesic Fields. Part I, Part II.

Notes and References

1. Book: Gelfand. I. M.. Israel Gelfand. Fomin. S. V.. Sergei Fomin. Calculus of variations . 2000. Dover Publications. Mineola, New York. 978-0486414485. 3. Unabridged repr.. Silverman. Richard A..
2. Book: Thiele, Rüdiger . Bradley . Robert E. . Sandifer . C. Edward . Leonhard Euler: Life, Work and Legacy . Elsevier . 2007 . 249 . Euler and the Calculus of Variations . https://books.google.com/books?id=75vJL_Y-PvsC&pg=PA249 . 9780080471297.
3. Book: Goldstine, Herman H. . 2012 . A History of the Calculus of Variations from the 17th through the 19th Century . Springer Science & Business Media . 110 . 9781461381068 . Herman Goldstine .
4. Book: van Brunt, Bruce . The Calculus of Variations . Springer . 2004 . 978-0-387-40247-5.
5. Ferguson . James . math/0402357 . Brief Survey of the History of the Calculus of Variations and its Applications . 2004 .
6. [Dimitri Bertsekas]
7. Bellman . Richard E. . Dynamic Programming and a new formalism in the calculus of variations . 1954 . Proc. Natl. Acad. Sci. . 4 . 231–235. 527981 . 16589462 . 40 . 10.1073/pnas.40.4.231. 1954PNAS...40..231B . free .
8. Web site: Richard E. Bellman Control Heritage Award . 2004 . American Automatic Control Council . 2013-07-28 . 2018-10-01 . https://web.archive.org/web/20181001032837/http://a2c2.org/awards/richard-e-bellman-control-heritage-award . dead .
9. Web site: Weisstein, Eric W. . Euler–Lagrange Differential Equation . mathworld.wolfram.com . Wolfram . Eq. (5).
10. Book: Kot, Mark . A First Course in the Calculus of Variations . American Mathematical Society . 2014 . 978-1-4704-1495-5 . Chapter 4: Basic Generalizations.
11. Manià. Bernard. 1934. Sopra un esempio di Lavrentieff. Bollenttino dell'Unione Matematica Italiana. 13. 147–153.
12. Ball & Mizel. 1985. One-dimensional Variational problems whose Minimizers do not satisfy the Euler-Lagrange equation.. Archive for Rational Mechanics and Analysis. 90. 4. 325–388. 10.1007/BF00276295. 1985ArRMA..90..325B. 55005550.
13. Ferriero. Alessandro. 2007. The Weak Repulsion property . Journal de Mathématiques Pures et Appliquées. 88. 4. 378–388. 10.1016/j.matpur.2007.06.002 .