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 number s. Functionals are often expressed as
definite integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ... s involving functions and their
derivative s. 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 ''
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ... s''. A related problem is posed by
Fermat's principle : light follows the path of shortest
optical length In optics, optical path length (OPL, denoted ''Λ'' in equations), also known as optical length or optical distance, is the product of the arc length, geometric length of the optical path followed by light and the refractive index of homogeneous med ... 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 problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ... s for the
Laplace equation satisfy the
Dirichlet's principle
In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation.
Formal statement
Dirichlet's principle states that, if the functi ... .
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).
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
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia[Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. ](_blank)Isaac Newton and Gottfried Leibniz also gave some early attention to the subject.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
Strauch, a German word meaning ''bush'' or ''shrub'', is a surname. Notable people with it include:
* Adolfo Strauch, (b. 1948), survivor of the Uruguayan Air Force Flight 571 crash
* Adolph Strauch (1822–1883), landscape architect
* Aegidius S ... 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
20 (twenty; Roman numeral XX) is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score.
In mathematics
*20 is a pronic number.
*20 is a tetrahedral number as 1, 4, 10, 20.
*20 is the ba ... 23rd Hilbert problem
Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the pro ... David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ... Oskar Bolza , Gilbert Ames Bliss , Emmy Noether , Leonida Tonelli
Leonida Tonelli (19 April 1885 – 12 March 1946) was an Italian mathematician, noted for creating Tonelli's theorem, a variation of Fubini's theorem, and for introducing semicontinuity methods as a common tool for the direct method in the ... Henri Lebesgue and Jacques Hadamard among others made significant contributions.Marston Morse applied calculus of variations in what is now called Morse theory .Lev Pontryagin , Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory .dynamic programming of Richard Bellman is an alternative to the calculus of variations. Extrema
The calculus of variations is concerned with the maxima or minima (collectively called extrema) of functionals. A functional maps functions to scalars
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
*Scalar (physics), a physical quantity that can be described by a single element of a number field such a ... y of a given function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ... domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ... J f if \Delta J = J sign
A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ... y in an arbitrarily small neighborhood of f. The function f is called an extremal function or extremal. The extremum J \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.converse may not hold. Finding strong extrema is more difficult than finding weak extrema.necessary condition that is used for finding weak extrema is the Euler–Lagrange equation . 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
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ... Euler–Lagrange equation .
Consider the functional
J
where
*x_1, x_2 are constants
Constant or The Constant may refer to:
Mathematics
* Constant (mathematics), a non-varying value
* Mathematical constant, a special number that arises naturally in mathematics, such as or
Other concepts
* Control variable or scientific const ... y(x) is twice continuously differentiable,
*y'(x) = \frac,
*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 local minimum at f, and \eta(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 + \varepsilon \eta \, .
The term \varepsilon \eta is called the variation of the function f and is denoted by \delta f. f + \varepsilon \eta for y in the functional J the result is a function of \varepsilon,
\Phi(\varepsilon) = J +\varepsilon\eta \, .
Since the functional J y = f the function \Phi(\varepsilon) has a minimum at \varepsilon = 0 and thus,
\Phi'(0) \equiv \left.\frac\_ = \int_^ \left.\frac\_ dx = 0 \, .
Taking the total derivative of L\left , y, y'\right where y = f + \varepsilon \eta and y' = f' + \varepsilon \eta' are considered as functions of \varepsilon rather than x, yields
\frac=\frac\frac + \frac\frac
and because \frac = \eta and \frac = \eta',
\frac=\frac\eta + \frac\eta'.
Therefore,
\begin
\int_^ \left.\frac\_ dx
& = \int_^ \left(\frac \eta + \frac \eta'\right)\, dx \\
& = \int_^ \frac \eta \, dx + \left.\frac \eta \_^ - \int_^ \eta \frac\frac \, dx \\
& = \int_^ \left(\frac \eta - \eta \frac\frac \right)\, dx\\
\end
where L\left , y, y'\right \to L\left , f, f'\right /math> when \varepsilon = 0 and we have used integration by parts on the second term. The second term on the second line vanishes because \eta = 0 at x_1 and x_2 by definition. Also, as previously mentioned the left side of the equation is zero so that
\int_^ \eta (x) \left(\frac - \frac\frac \right) \, dx = 0 \, .
According to the fundamental lemma of calculus of variations , the part of the integrand in parentheses is zero, i.e.
\frac -\frac \frac=0
which is called the Euler–Lagrange equation. The left hand side of this equation is called the functional derivative
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ... J \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 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
with
y'(x) = \frac \, , \ \ y_1=f(x_1) \, , \ \ y_2=f(x_2) \, .
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
\frac -\frac \frac=0
with
L = \sqrt \, .
Since f does not appear explicitly in L, the first term in the Euler–Lagrange equation vanishes for all f(x) and thus,
\frac \frac = 0 \, .
Substituting for L and taking the derivative,
\frac \ \frac \ = 0 \, .
Thus
\frac = c \, ,
for some constant c. Then
\frac = c^2 \, ,
where
0 \le c^2<1.
Solving, we get
'(x)
The apostrophe ( or ) is a punctuation mark, and sometimes a diacritical mark, in languages that use the Latin alphabet and some other alphabets. In English, the apostrophe is used for two basic purposes:
* The marking of the omission of one o ... f'(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 \qquad \text \ \ m = \frac \quad \text \quad b = \frac
and we have thus found the extremal function f(x) that minimizes the functional A A 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 \frac = 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
L - f' \frac = C \, ,
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 \frac = 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 = \int_^ f(x, y(x), y'(x), \dots, y^(x)) dx, then y must satisfy the Euler– Poisson equation, \frac - \frac \left( \frac \right) + \dots + (-1)^ \frac \left \frac \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
\frac \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:
L
= \.
Clearly, x(t) = t^ minimizes the functional, but we find any function x \in W^ gives a value bounded away from the infimum.
Examples (in one-dimension) are traditionally manifested across W^ and W^, but Ball and Mizel procured the first functional that displayed Lavrentiev's Phenomenon across W^ and W^ for 1 \leq p < q < \infty. 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. 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 varphi = \iint_D \sqrt \,dx\,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:
\varphi_(1 + \varphi_y^2) + \varphi_(1 + \varphi_x^2) - 2\varphi_x \varphi_y \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 varphi = \frac\iint_D \nabla \varphi \cdot \nabla \varphi \, dx\, 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 + \varepsilon v /math> must vanish:
\left.\frac V + \varepsilon v _ = \iint_D \nabla u \cdot \nabla v \, dx\,dy = 0.
Provided that u has two derivatives, we may apply the divergence theorem to obtain
\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 \, 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
\iint_D v\nabla \cdot \nabla u \,dx\,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
\nabla \cdot \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 varphi = \int_^ (x\varphi')^2 \, 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 equation
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form
:Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\,
wher ... Generalization to other boundary value problems
A more general expression for the potential energy of a membrane is
V varphi = \iint_D \left \frac \nabla \varphi \cdot \nabla \varphi + f(x,y) \varphi \right \, dx\,dy \, + \int_C \left \frac \sigma(s) \varphi^2 + g(s) \varphi \right \, 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 + \varepsilon v /math> is given by
\iint_D \left \nabla u \cdot \nabla v + f v \right \, dx\, dy + \int_C \left \sigma u v + g v \right \, ds = 0.
If we apply the divergence theorem, the result is
\iint_D \left -v \nabla \cdot \nabla u + v f \right \, dx \, dy + \int_C v \left \frac + \sigma u + g \right \, ds =0.
If we first set v = 0 on C, the boundary integral vanishes, and we conclude as before that
- \nabla \cdot \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
\frac + \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 \iint_D f \, dx\,dy + \int_C g \, ds \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
\iint_D f \, dx\,dy + \int_C g \, 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
The Sturm–Liouville eigenvalue problem involves a general quadratic form
Q varphi = \int_^ \left p(x) \varphi'(x)^2 + q(x) \varphi(x)^2 \right \, dx,
where \varphi is restricted to functions that satisfy the boundary conditions
\varphi(x_1)=0, \quad \varphi(x_2)=0.
Let R be a normalization integral
R varphi =\int_^ r(x)\varphi(x)^2 \, 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 \varphi satisfying the endpoint conditions. It is shown below that the Euler–Lagrange equation for the minimizing u is
-(p u')' +q u -\lambda r u = 0,
where \lambda is the quotient
\lambda = \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 \lambda will be denoted by \lambda_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
\int_^ r(x) u_1(x) \varphi(x) \, 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 \varphi vanish at the endpoints, we may not impose any condition at the endpoints, and set
Q varphi = \int_^ \left p(x) \varphi'(x)^2 + q(x)\varphi(x)^2 \right \, dx + a_1 \varphi(x_1)^2 + a_2 \varphi(x_2)^2,
where a_1 and a_2 are arbitrary. If we set \varphi = u + \varepsilon v the first variation for the ratio Q/R is
V_1 = \frac \left( \int_^ \left p(x) u'(x)v'(x) + q(x)u(x)v(x) -\lambda r(x) u(x) v(x) \right \, dx + a_1 u(x_1)v(x_1) + a_2 u(x_2)v(x_2) \right),
where λ is given by the ratio Q \frac V_1 = \int_^ 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) (x_2) u'(x_2) + a_2 u(x_2)
X, or x, is the twenty-fourth and third-to-last letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''"ex"'' (pronounced ), ...
If we first require that v vanish at the endpoints, the first variation will vanish for all such v only if
-(p u')' + q u -\lambda r u =0 \quad \hbox \quad x_1 < x < x_2.
If u satisfies this condition, then the first variation will vanish for arbitrary v only if
-p(x_1)u'(x_1) + a_1 u(x_1)=0, \quad \hbox \quad p(x_2) u'(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 varphi = \iiint_D p(X) \nabla \varphi \cdot \nabla \varphi + q(X) \varphi^2 \, dx \, dy \, dz + \iint_B \sigma(S) \varphi^2 \, dS,
and
R varphi = \iiint_D r(X) \varphi(X)^2 \, dx \, dy \, 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
-\nabla \cdot (p(X) \nabla u) + q(x) u - \lambda r(x) u=0,
where
\lambda = \frac.
The minimizing u must also satisfy the natural boundary condition
p(S) \frac + \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
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 In applied mathematics and the calculus of variations, the first variation of a functional ''J''(''y'') is defined as the linear functional \delta J(y) mapping the function ''h'' to
:\delta J(y,h) = \lim_ \frac = \left.\frac J(y + \varepsilon ... A (the derivative of A with respect to ε) is
\delta A _0,f_1 = \int_^ \left \frac + n_y (x,f_0) f_1 \sqrt \right dx.
After integration by parts of the first term within brackets, we obtain the Euler–Lagrange equation
-\frac \left frac \right + n_y (x,f_0) \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. Let
n(x,y) = \begin
n_ & \text \quad x<0, \\
n_ & \text \quad x>0,
\end
where 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
\delta A _0,f_1 = f_1(0)\left n_\frac - n_\frac \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
Snell's law (also known as Snell–Descartes law and ibn-Sahl law and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through ... 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 \dot X(t) be its tangent vector. The optical length of the curve is given by
A
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
\frac P = \sqrt \, \nabla n,
where
P = \frac.
It follows from the definition that P satisfies
P \cdot P = n(X)^2.
Therefore, the integral may also be written as
A
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 \nabla \cdot \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
\varphi_t^2 = c(X)^2 \, \nabla \varphi \cdot \nabla \varphi.
We may look for solutions in the form
\varphi(t,X) = t - \psi(X).
In that case, \psi satisfies
\nabla \psi \cdot \nabla \psi = n^2,
where n=1/c. According to the theory of first-order partial differential equation s, if P = \nabla \psi, then P satisfies
\frac = n \, \nabla n,
along a system of curves (the light rays) that are given by
\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
\frac = \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
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
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potentia ... 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 = \int_^ L(x, \dot x, t) \, 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:
\frac \frac = \frac,
and they are equivalent to Newton's equations of motion (for such systems).
The conjugate momenta P are defined by
p = \frac.
For example, if
T = \frac m \dot x^2,
then p = m \dot x.
Hamiltonian mechanics results if the conjugate momenta are introduced in place of \dot x by a Legendre transformation of the Lagrangian L into the Hamiltonian H defined by
H(x, p, t) = p \,\dot x - L(x,\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 :
\frac + H\left(x,\frac,t\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
A tautochrone or isochrone curve (from Greek prefixes tauto- meaning ''same'' or iso- ''equal'', and chrono ''time'') is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independe ... isoperimetric problems
* Calculating geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ... minimal surface s and solving Plateau's problem
* Optimal control
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and ... Analytical mechanics
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the ... Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ... 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
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ... 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.J 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
\Delta J +h - J
The functional J \Delta J
where \varphi \, h\, is the norm of h, and \varepsilon \to 0 as \, h\, \to 0. The linear functional \varphi J \delta J
The functional J \Delta J
where \varphi_1 \varphi_2 \varepsilon \to 0 as \, h\, \to 0. The quadratic functional \varphi_2 J \delta^2 J
The second variation \delta^2 J \delta^2J
for all h and for some constant k > 0 . See also
* First variation In applied mathematics and the calculus of variations, the first variation of a functional ''J''(''y'') is defined as the linear functional \delta J(y) mapping the function ''h'' to
:\delta J(y,h) = \lim_ \frac = \left.\frac J(y + \varepsilon ... Isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ... 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 In mathematics, the inverse problem for Lagrangian mechanics is the problem of determining whether a given system of ordinary differential equations can arise as the Euler–Lagrange equations for some Lagrangian function.
There has been a gr ... Obstacle problem The obstacle problem is a classic motivating example in the mathematical study of variational inequalities and free boundary problems. The problem is to find the equilibrium position of an elastic membrane whose boundary is held fixed, and which ... Perturbation methods
* Young measure
In mathematical analysis, a Young measure is a parameterized measure that is associated with certain subsequences of a given bounded sequence of measurable functions. They are a quantification of the oscillation effect of the sequence in the limi ... Optimal control
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and ... Direct method in calculus of variations
In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies ... Noether's theorem
* De Donder–Weyl theory In mathematical physics, the De Donder–Weyl theory is a generalization of the Hamiltonian formalism in the calculus of variations and classical field theory over spacetime which treats the space and time coordinates on equal footing. In this frame ... Variational Bayesian methods
* Chaplygin problem
* Nehari manifold
In the calculus of variations, a branch of mathematics, a Nehari manifold is a manifold of functions, whose definition is motivated by the work of . It is a differentiable manifold associated to the Dirichlet problem for the semilinear ellipt ... Hu–Washizu principle
* Luke's variational principle
In fluid dynamics, Luke's variational principle is a Lagrangian variational description of the motion of surface waves on a fluid with a free surface, under the action of gravity. This principle is named after J.C. Luke, who published it in 1967 ... Mountain pass theorem The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz. Given certain conditions on a function, the theorem demonstrates the existence of a saddle point. The th ... Measures of central tendency as solutions to variational problems
* Stampacchia Medal The Stampacchia Gold Medal is an international prize awarded every three years by the Italian Mathematical Union (''Unione Matematica Italiana'' - ''UMI'' ) together with the Ettore Majorana Foundation (Erice), in recognition of outstanding contrib ... Fermat Prize
* Convenient vector space In mathematics, convenient vector spaces are locally convex vector spaces satisfying a very mild completeness condition.
Traditional differential calculus is effective in the analysis of finite-dimensional vector spaces and for Banach spaces. B ... Notes
References
Further reading
* Benesova, B. and Kruzik, M."Weak Lower Semicontinuity of Integral Functionals and Applications" Bolza, O. Lectures on the Calculus of Variations Variational Methods with Applications in Science and Engineering Courant, R. Dirichlet's principle, conformal mapping and minimal surfaces Dacorogna, Bernard :Introduction Introduction to the Calculus of Variations Calculus of Variations Calculus of Variations Calculus of Variations The Calculus of Variations and Functional Analysis with Optimal Control and Applications in Mechanics Applied Mathematics Calculus of variations Mathematical Tools for Physicists Calculus of Variations with Applications to Physics and Engineering External links
Variational calculus Encyclopedia of Mathematics ''.
calculus of variations PlanetMath ''.
Calculus of Variations MathWorld ''.
Calculus of variations Mathematics - Calculus of Variations and Integral Equations YouTube .
* Selected papers on Geodesic FieldsPart I Part II
Optimization in vector spaces
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