The calculus of variations (or Variational Calculus) is a field of
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ... 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
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ... 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 ... s involving functions and their
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ... s. Functions that maximize or minimize functionals may be found using the
Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ... 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
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ... 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 connecti ... s''. A related problem is posed by
Fermat's principle
Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the pat ... : light follows the path of shortest
optical length connecting two points, which depends upon the material of the medium. One corresponding concept in
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objec ... 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
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \n ... 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 funct ... .
Plateau's problem
In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The 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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ... .
History
The calculus of variations may be said to begin with
Newton's minimal resistance problem in 1687, followed by the
brachistochrone curve
In physics and mathematics, a brachistochrone curve (), or curve of fastest descent, is the one lying on the plane between a point ''A'' and a lower point ''B'', where ''B'' is not directly below ''A'', on which a bead slides frictionlessly un ... problem raised by
Johann Bernoulli
Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating Le ... (1696).
It immediately occupied the attention of
Jakob Bernoulli and the
Marquis de l'Hôpital , but
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ... 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
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ... Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ... Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ... Siméon Poisson (1831), Mikhail Ostrogradsky
Mikhail Vasilyevich Ostrogradsky (transcribed also ''Ostrogradskiy'', Ostrogradskiĭ) (russian: Михаи́л Васи́льевич Острогра́дский, ua, Миха́йло Васи́льович Острогра́дський; 24 Sep ... Carl Jacobi (1837) have been among the contributors. An important general work is that of Sarrus (1842) which was condensed and improved by Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ... 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 ... Jellett (1850), Otto Hesse
Ludwig Otto Hesse (22 April 1811 – 4 August 1874) was a German mathematician. Hesse was born in Königsberg, Prussia, and died in Munich, Bavaria. He worked mainly on algebraic invariants, and geometry. The Hessian matrix, the Hesse nor ... Alfred Clebsch
Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. ... Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ... 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 published in 1900 encouraged further development.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 ... Oskar Bolza
Oskar Bolza (12 May 1857 – 5 July 1942) was a German mathematician, and student of Felix Klein. He was born in Bad Bergzabern, Palatinate (region), Palatinate, then a district of Bavaria, known for his research in the calculus of variations, p ... Gilbert Ames Bliss
Gilbert Ames Bliss, (9 May 1876 – 8 May 1951), was an American mathematician, known for his work on the calculus of variations.
Life
Bliss grew up in a Chicago family that eventually became affluent; in 1907, his father became president of the ... Emmy Noether
Amalie Emmy NoetherEmmy is the '' Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ... Leonida Tonelli
Leonida Tonelli (19 April 1885 – 12 March 1946) was an Italian people, Italian mathematician, noted for creating Fubini's theorem#Tonelli's theorem for non-negative measurable functions, Tonelli's theorem, a variation of Fubini's theorem, and f ... Henri Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ... Jacques Hadamard
Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations.
Biography
The son of a teac ... Marston Morse applied calculus of variations in what is now called Morse theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ... Lev Pontryagin
Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician. He was born in Moscow and lost his eyesight completely d ... Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ... dynamic programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.
... Richard Bellman
Richard Ernest Bellman (August 26, 1920 – March 19, 1984) was an American applied mathematician, who introduced dynamic programming in 1953, and made important contributions in other fields of mathematics, such as biomathematics. He founde ... 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 ... 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
* ... 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 ... 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
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth o ... Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ... 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 w ... Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ... 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
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ... 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
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivat ... \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
In mathematics, specifically in the calculus of variations, a variation of a function can be concentrated on an arbitrarily small interval, but not a single point.
Accordingly, the necessary condition of extremum (functional derivative equal ze ... \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 w ... J \delta J/\delta f(x).
In general this gives a second-order ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ... f(x). The Euler–Lagrange equation is a necessary , but not sufficient
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ... 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
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ... 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) 2=\frac
which implies that
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
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ... 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
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ... 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
In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem ... 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 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 functio ... Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ... 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,\,
whe ... 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 The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, originated in the context of solving physical boundary value problems and named after Lord Rayleigh and Walther Ritz.
The name Rayleigh–Ritz is being debate ... 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)
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
Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the pat ... 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 h ... 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 Hamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrical ... Hamiltonian optics Hamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrical ... 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 throug ... 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 Hamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrical ... Hamiltonian optics Hamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrical ... = Connection with the wave equation
=
The wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and ... 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 In mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of ''n'' variables. The equation takes the form
: F(x_1,\ldots,x_n,u,u_,\ldots u_) =0. \,
... 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
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ... 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 potenti ... Hamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function, ... 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
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ... \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
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mecha ... \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
In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field.
The catenary curve has a U-like shape, superfici ... 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 indepen ... isoperimetric
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 by ... 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 connecti ... minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces tha ... Plateau's problem
In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The 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 and Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ... Hamiltonian optics Hamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrical ... Variational method (quantum mechanics) In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. This allows calculating approximate wavefunctions such as molecular orbitals. The basis for t ... Variational Bayesian methods
Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning. They are typically used in complex statistical models consisting of observed variables (usually ... Variational methods in general relativity
Variational methods in general relativity refers to various mathematical techniques that employ the use of variational calculus in Einstein's theory of general relativity. The most commonly used tools are Lagrangians and Hamiltonians and are used ... Finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ... Total variation denoising
In signal processing, particularly image processing, total variation denoising, also known as total variation regularization or total variation filtering, is a noise removal process ( filter). It is based on the principle that signals with excessi ... 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-dimensio ... 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 h ... 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
In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those funct ... Variational bicomplex
In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations. For instance, this is the case of classical field theory on fiber bun ... Fermat's principle
Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the pat ... Principle of least action
The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the '' action'' of a mechanical system, yields the equations of motion for that system. The principle states tha ... Infinite-dimensional optimization In certain optimization problems the unknown optimal solution might not be a number or a vector, but rather a continuous quantity, for example a function or the shape of a body. Such a problem is an infinite-dimensional optimization problem, becaus ... Finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ... Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ... Ekeland's variational principle In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exist nearly optimal solutions to some optimization problems.
Ekeland's principle can be used when the lower level set of a ... Inverse problem for Lagrangian mechanics
* 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 i ... 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 reli ... Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ... De Donder–Weyl theory
* Variational Bayesian methods
Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning. They are typically used in complex statistical models consisting of observed variables (usually ... Chaplygin problem
In mathematics, particularly in the fields of nonlinear dynamics and the calculus of variations, the Chaplygin problem is an isoperimetric problem with a differential constraint. Specifically, the problem is to determine what flight path an airp ... Nehari manifold
* Hu–Washizu principle
* Luke's variational principle
* 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. ... 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
The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structu ... calculus of variations PlanetMath
PlanetMath is a free, collaborative, mathematics online encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be c ... Calculus of Variations MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Di ... Calculus of variations Mathematics - Calculus of Variations and Integral Equations YouTube
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Optimization in vector spaces
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