The calculus of variations (or variational calculus) is a field of
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ... 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ... s. Functionals are often expressed as
definite integral s involving functions and their
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ... 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 locally 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 conn ... s''. A related problem is posed by
Fermat's principle : light follows the path of shortest
optical length connecting two points, which depends upon the material of the medium. One corresponding concept in
mechanics
Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ... is the
principle of least/stationary action .
Many important problems involve functions of several variables. Solutions of
boundary value problem s for the
Laplace equation satisfy the
Dirichlet's principle .
Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in soapy water. Although such experiments are relatively easy to perform, their mathematical formulation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ... .
History
The calculus of variations began with the work of
Isaac Newton
Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ... , such as with
Newton's minimal resistance problem , which he formulated and solved in 1685, and later published in his ''
Principia '' in 1687,
which was the first problem in the field to be formulated and correctly solved,
and was also one of the most difficult problems tackled by variational methods prior to the twentieth century.
This problem was followed by the
brachistochrone curve problem raised by
Johann Bernoulli
Johann Bernoulli (also known as Jean in French or John in English; – 1 January 1748) was a Swiss people, Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infin ... (1696),
which was similar to one raised by
Galileo Galilei
Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642), commonly referred to as Galileo Galilei ( , , ) or mononymously as Galileo, was an Italian astronomer, physicist and engineer, sometimes described as a poly ... in 1638, but he did not solve the problem explicity nor did he use the methods based on calculus.
Bernoulli had solved the problem, using the principle of least time in the process, but not calculus of variations, whereas Newton did to solve the problem in 1697, and as a result, he pioneered the field with his work on the two problems.
The problem would immediately occupy the attention of
Jacob Bernoulli
Jacob Bernoulli (also known as James in English or Jacques in French; – 16 August 1705) was a Swiss mathematician. He sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy and was an early proponent of Leibniz ... and the
Marquis de l'Hôpital , but
Leonhard Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ... first elaborated the subject, beginning in 1733.
Joseph-Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia[Adrien-Marie Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. ](_blank)Isaac Newton
Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ... Gottfried Leibniz
Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Isaac Newton, Sir Isaac Newton, with the creation of calculus in ad ... Vincenzo Brunacci (1810), Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ... Siméon Poisson (1831), Mikhail Ostrogradsky (1834), and Carl Jacobi (1837) have been among the contributors. An important general work is that of Pierre Frédéric Sarrus (1842) which was condensed and improved by Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ... Strauch (1849), John Hewitt Jellett (1850), Otto Hesse (1857), 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 Humboldt ... Karl Weierstrass
Karl Theodor Wilhelm 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 and trained as a school t ... 20th and the 23rd Hilbert problem published in 1900 encouraged further development.David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad range of fundamental idea ... 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, then a district of Bavaria, known for his research in the calculus of variations, particularly influen ... Gilbert Ames Bliss , Emmy Noether
Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ... Leonida Tonelli
Leonida Tonelli (19 April 1885 – 12 March 1946) was an Italian people, Italian mathematician, noted for proving Fubini's theorem#Tonelli's theorem for non-negative measurable functions, Tonelli's theorem, a variation of Fubini's theorem, and f ... 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 , so functionals have been described as "functions of functions." Functionals have extrema with respect to the elements 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 ve ... domain . A functional J f if \Delta J = J sign for all 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 is equal to zero. This leads to solving the associated Euler–Lagrange equation .
Consider the functional
J
where
*x_1, x_2 are constants ,
*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
In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with res ... 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 antiderivati ... \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 of J \delta J or \delta f(x).
In general this gives a second-order ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ... 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
Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ... 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 = mx+b. 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
Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems (see Noether's second theorem) published by the mat ... 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
Johann Peter Gustav Lejeune Dirichlet (; ; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In analysis, he advanced the theory o ... 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 s; see Jost and Li–Jost (1998).
Generalization to other boundary value problems
A more general expression for the potential energy of a membrane is
V 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
In linear algebra, an eigenvector ( ) or characteristic vector is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an e ... Q p(x) y'(x)^2 + q(x) y(x)^2 \right \, dx,
where y is restricted to functions that satisfy the boundary conditions
y(x_1)=0, \quad y(x_2)=0.
Let R be a normalization integral
R
The functions p(x) and r(x) are required to be everywhere positive and bounded away from zero. The primary variational problem is to minimize the ratio Q/R among all y satisfying the endpoint conditions, which is equivalent to minimizing Q R 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) y(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 y vanish at the endpoints, we may not impose any condition at the endpoints, and set
Q p(x) y'(x)^2 + q(x)y(x)^2 \right \, dx + a_1 y(x_1)^2 + a_2 y(x_2)^2,
where a_1 and a_2 are arbitrary. If we set y = u + \varepsilon v , the first variation for the ratio Q/R is
V_1 = \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 \lambda 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 /math>,
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 of A (the derivative of A with respect to \varepsilon ) 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 for refraction requires that these terms be equal. As this calculation demonstrates, Snell's law is equivalent to vanishing of the first variation of the optical path length.
Fermat's principle in three dimensions
It is expedient to use vector notation: let X = (x_1,x_2,x_3), let t be a parameter, let X(t) be the parametric representation of a curve C, and let \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
In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion.
In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ... U its potential energy
In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ... 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
In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ... \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 wire rope, 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, ... Newton's minimal resistance problem
* Solution to the brachistochrone problem
* Solution to the tautochrone problem
* Solution to isoperimetric problems
* Calculating geodesic
In geometry, a geodesic () is a curve representing in some sense the locally 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 conn ... minimal surface s and solving Plateau's problem
* Optimal control
* Analytical mechanics
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related formulations of classical mechanics. Analytical mechanics uses '' scalar'' properties of motion representing the sy ... Lagrangian and Hamiltonian mechanics
In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ... 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
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 tran ... 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 excess ... image processing
An image or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a pr ... 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
* Isoperimetric inequality
* Variational principle
* Variational bicomplex
* Fermat's principle
* Principle of least action
* Infinite-dimensional optimization
* Finite element method
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 tran ... 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 (for example, Inner product space#Definition, inner product, Norm (mathematics ... Ekeland's variational principle
* Inverse problem for Lagrangian mechanics
* Obstacle problem
* Perturbation methods
* Young measure
In mathematical analysis, a Young measure is a parameterized measure (mathematics), measure that is associated with certain subsequences of a given bounded sequence of measurable functions. They are a quantification of the oscillation effect of th ... Optimal control
* Direct method in calculus of variations
* Noether's theorem
Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems (see Noether's second theorem) published by the mat ... De Donder–Weyl theory
* Variational Bayesian methods
* Chaplygin problem
* Nehari manifold
* Hu–Washizu principle
* Luke's variational principle
* Mountain pass theorem
*
* Measures of central tendency as solutions to variational problems
* Stampacchia Medal
* Fermat Prize
* Convenient vector space
* Variational vector field
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
PlanetMath is a free content, free, collaborative, mathematics online encyclopedia. Intended to be comprehensive, the project is currently hosted by the University of Waterloo. The site is owned by a US-based nonprofit corporation, "PlanetMath.org ... Calculus of Variations MathWorld ''.
Calculus of variations Mathematics - Calculus of Variations and Integral Equations YouTube
YouTube is an American social media and online video sharing platform owned by Google. YouTube was founded on February 14, 2005, by Steve Chen, Chad Hurley, and Jawed Karim who were three former employees of PayPal. Headquartered in ... Part I Part II
Optimization in vector spaces
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