Английская Википедия:Calculus of variations

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Шаблон:Short description Шаблон:Redirect Шаблон:Calculus

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.Шаблон:Efn 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 Jakob 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]Шаблон:Efn

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.[4]

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.[4] 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.[5] The dynamic programming of Richard Bellman is an alternative to the calculus of variations.[6][7][8]Шаблон:Efn

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 <math>y</math> of a given function space defined over a given 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 for all <math>y</math> in an arbitrarily small neighborhood of <math>f.</math>Шаблон:Efn The function <math>f</math> is called an extremal function or extremal.Шаблон:Efn 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 strong extrema or weak extrema, depending on whether the first derivatives of the continuous functions are respectively all continuous or not.[9]

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.[10] An example of a necessary condition that is used for finding weak extrema is the Euler–Lagrange equation.[11]Шаблон:Efn

Euler–Lagrange equation

Шаблон:Main 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.Шаблон:Efn

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 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>[12]Шаблон:Efn

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,Шаблон:Efn <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/\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, but not 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.

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 Шаблон:Mvar is a function of Шаблон:Mvar 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 = f(x).</math> In other words, the shortest distance between two points is a straight line.Шаблон:Efn

Beltrami's identity

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'(x)</math> but <math>x</math> does not appear separately. In that case, the Euler–Lagrange equation can be simplified to the Beltrami identity[13] <math display="block">L - f' \frac{\partial L}{\partial f'} = 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'(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.

Euler–Poisson equation

If <math>S</math> depends on higher-derivatives of <math>y(x),</math> that is, if <math display="block">S = \int_{a}^{b} f(x, y(x), y'(x), \dots, y^{(n)}(x)) dx,</math> then <math>y</math> must satisfy the Euler–Poisson equation,[14] <math display="block">\frac{\partial f}{\partial y} - \frac{d}{dx} \left( \frac{\partial f}{\partial y'} \right) + \dots + (-1)^{n} \frac{d^n}{dx^n} \left[ \frac{\partial f}{\partial y^{(n)}} \right]= 0.</math>

Du Bois-Reymond's theorem

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 display="block">\frac{\partial^2 L}{\partial f'^2} \ne 0,</math> then <math>f</math> 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:[15] <math display="block">L[x] = \int_0^1 (x^3-t)^2 x'^6,</math> <math display="block">{A} = \{x \in W^{1,1}(0,1) : x(0)=0,\ x(1)=1\}.</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[16] 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.

Connected with the Lavrentiev Phenomenon is the repulsion property: any functional displaying Lavrentiev's Phenomenon will display the weak repulsion property.[17]

Functions of several variables

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 display="block">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 display="block">\varphi_{xx}(1 + \varphi_y^2) + \varphi_{yy}(1 + \varphi_x^2) - 2\varphi_x \varphi_y \varphi_{xy} = 0.</math> 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 <math display="block">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 display="block">\left.\frac{d}{d\varepsilon} V[u + \varepsilon v]\right|_{\varepsilon=0} = \iint_D \nabla u \cdot \nabla v \, dx\,dy = 0.</math> Provided that u has two derivatives, we may apply the divergence theorem to obtain <math display="block">\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,</math> 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 display="block">\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 display="block">\nabla \cdot \nabla u= 0 </math>in <math>D.</math>

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 <math display="block">W[\varphi] = \int_{-1}^{1} (x\varphi')^2 \, dx</math> among all functions <math>\varphi</math> that satisfy <math>\varphi(-1)=-1</math> and <math>\varphi(1)=1.</math> <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>Шаблон:Efn 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 <math display="block">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>V[u + \varepsilon v]</math> is given by <math display="block">\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> If we apply the divergence theorem, the result is <math display="block">\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 display="block">- \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 display="block">\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 display="block">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 display="block">\iint_D f \, dx\,dy + \int_C g \, ds =0.</math> 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

Шаблон:See also The Sturm–Liouville eigenvalue problem involves a general quadratic form <math display="block">Q[y] = \int_{x_1}^{x_2} \left[ p(x) y'(x)^2 + q(x) y(x)^2 \right] \, dx, </math> where <math>y</math>is restricted to functions that satisfy the boundary conditions <math display="block">y(x_1)=0, \quad y(x_2)=0. </math> Let <math>R</math> be a normalization integral <math display="block">R[y] =\int_{x_1}^{x_2} r(x)y(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>y</math> satisfying the endpoint conditions, which is equivalent to minimizing <math>Q[y]</math> under the constraint that <math>R[y]</math> is constant. It is shown below that the Euler–Lagrange equation for the minimizing <math>u</math> is <math display="block">-(p u')' +q u -\lambda r u = 0, </math> where <math>\lambda</math> is the quotient <math display="block">\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 display="block">\int_{x_1}^{x_2} r(x) u_1(x) y(x) \, dx = 0. </math> 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 <math>y</math> vanish at the endpoints, we may not impose any condition at the endpoints, and set <math display="block">Q[y] = \int_{x_1}^{x_2} \left[ p(x) y'(x)^2 + q(x)y(x)^2 \right] \, dx + a_1 y(x_1)^2 + a_2 y(x_2)^2, </math> where <math>a_1</math> and <math>a_2</math> are arbitrary. If we set <math>y = u + \varepsilon v</math>the first variation for the ratio <math>Q/R</math> is <math display="block">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), </math> where λ is given by the ratio <math>Q[u]/R[u]</math> as previously. After integration by parts, <math display="block">\frac{R[u]}{2} V_1 = \int_{x_1}^{x_2} v(x) \left[ -(p u')' + q u -\lambda r u \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)]. </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 display="block">-(p u')' + 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 display="block">-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.</math> 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 <math>D</math> with boundary <math>B</math> in three dimensions we may define <math display="block">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> and <math display="block">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 display="block">-\nabla \cdot (p(X) \nabla u) + q(x) u - \lambda r(x) u=0,</math> where <math display="block">\lambda = \frac{Q[u]}{R[u]}.</math> The minimizing <math>u</math> must also satisfy the natural boundary condition <math display="block">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).

Applications

Optics

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 display="block">A[f] = \int_{x_0}^{x_1} n(x,f(x)) \sqrt{1 + f'(x)^2} dx, </math> where the refractive index <math>n(x,y)</math> depends upon the material. 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 display="block">\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.</math>

After integration by parts of the first term within brackets, we obtain the Euler–Lagrange equation <math display="block">-\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. </math>

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. Let <math display="block">n(x,y) = \begin{cases} n_{(-)} & \text{if} \quad x<0, \\ n_{(+)} & \text{if} \quad x>0, \end{cases}</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'</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 display="block">\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].</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.

Fermat's principle in three dimensions

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 display="block">A[C] = \int_{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 display="block">\frac{d}{dt} P = \sqrt{ \dot X \cdot \dot X} \, \nabla n, </math> where <math display="block">P = \frac{n(X) \dot X}{\sqrt{\dot X \cdot \dot X} }.</math>

It follows from the definition that <math>P</math> satisfies <math display="block">P \cdot P = n(X)^2. </math>

Therefore, the integral may also be written as <math display="block">A[C] = \int_{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.

Connection with the wave equation

The wave equation for an inhomogeneous medium is <math display="block">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 display="block">\varphi_t^2 = c(X)^2 \, \nabla \varphi \cdot \nabla \varphi. </math>

We may look for solutions in the form <math display="block">\varphi(t,X) = t - \psi(X). </math>

In that case, <math>\psi</math> satisfies <math display="block">\nabla \psi \cdot \nabla \psi = n^2, </math> where <math>n=1/c.</math> According to the theory of first-order partial differential equations, if <math>P = \nabla \psi,</math> then <math>P</math> satisfies <math display="block">\frac{dP}{ds} = n \, \nabla n,</math> along a system of curves (the light rays) that are given by <math display="block">\frac{dX}{ds} = P. </math>

These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification <math display="block">\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.

Mechanics

Шаблон:Main 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 display="block">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 display="block">S = \int_{t_0}^{t_1} L(x, \dot x, t) \, dt</math> is stationary with respect to variations in the path <math>x(t).</math> The Euler–Lagrange equations for this system are known as Lagrange's equations: <math display="block">\frac{d}{dt} \frac{\partial L}{\partial \dot x} = \frac{\partial L}{\partial x}, </math> and they are equivalent to Newton's equations of motion (for such systems).

The conjugate momenta <math>P</math> are defined by <math display="block">p = \frac{\partial L}{\partial \dot x}. </math> For example, if <math display="block">T = \frac{1}{2} m \dot x^2, </math> then <math display="block">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 display="block">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 display="block">\frac{\partial \psi}{\partial t} + H\left(x,\frac{\partial \psi}{\partial x},t\right) = 0.</math>

Further applications

Further applications of the calculus of variations include the following:

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 variationШаблон:Efn is defined as the linear part of the change in the functional, and the second variationШаблон:Efn is defined as the quadratic part.[18]

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 isШаблон:Efn <math display="block">\Delta J[h] = J[y+h] - J[y].</math>

The functional <math>J[y]</math> is said to be differentiable if <math display="block">\Delta J[h] = \varphi [h] + \varepsilon \|h\|,</math> where <math>\varphi[h]</math> is a linear functional,Шаблон:Efn <math>\|h\|</math> is the norm of <math>h,</math>Шаблон:Efn 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,[19] <math display="block">\delta J[h] = \varphi[h].</math>

The functional <math>J[y]</math> is said to be twice differentiable if <math display="block">\Delta J[h] = \varphi_1 [h] + \varphi_2 [h] + \varepsilon \|h\|^2,</math> where <math>\varphi_1[h]</math> is a linear functional (the first variation), <math>\varphi_2[h]</math> is a quadratic functional,Шаблон:Efn and <math>\varepsilon \to 0</math> as <math>\|h\| \to 0.</math> The quadratic functional <math>\varphi_2[h]</math> is the second variation of <math>J[y]</math> and is denoted by,[20] <math display="block">\delta^2 J[h] = \varphi_2[h].</math>

The second variation <math>\delta^2 J[h]</math> is said to be strongly positive if <math display="block">\delta^2J[h] \ge k \|h\|^2,</math> for all <math>h</math> and for some constant <math>k > 0</math>.[21]

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.

Шаблон:Quote box

See also

Шаблон:Div col

Шаблон:Div col end

Notes

Шаблон:Notelist

References

Шаблон:Reflist

Further reading

External links

Шаблон:Analysis in topological vector spaces Шаблон:Convex analysis and variational analysis

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  1. ↑ Шаблон:Cite book
  2. ↑ Шаблон:Cite book
  3. ↑ Шаблон:Cite book
  4. ↑ 4,0 4,1 4,2 Шаблон:Cite book
  5. ↑ 5,0 5,1 Шаблон:Cite arXiv
  6. ↑ Dimitri Bertsekas. Dynamic programming and optimal control. Athena Scientific, 2005.
  7. ↑ Шаблон:Cite journal
  8. ↑ Шаблон:Cite web
  9. ↑ Шаблон:Harvnb
  10. ↑ Шаблон:Harvnb
  11. ↑ Шаблон:Harvnb
  12. ↑ Ошибка цитирования Неверный тег <ref>; для сносок CourHilb1953P184 не указан текст
  13. ↑ Шаблон:Cite web
  14. ↑ Шаблон:Cite book
  15. ↑ Шаблон:Cite journal
  16. ↑ Шаблон:Cite journal
  17. ↑ Шаблон:Cite journal
  18. ↑ Шаблон:Harvnb
  19. ↑ Шаблон:Harvnb
  20. ↑ Шаблон:Harvnb
  21. ↑ Шаблон:Harvnb
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