A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends upon the material of the medium. One corresponding concept in mechanics is the principle of least/stationary action.
Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet's principle. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in soapy water. Although such experiments are relatively easy to perform, their mathematical formulation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.
The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696).[2] It immediately occupied the attention of Jacob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733. Lagrange was influenced by Euler's work to contribute significantly to the theory. After Euler saw the 1755 work of the 19-year-old Lagrange, Euler dropped his own partly geometric approach in favor of Lagrange's purely analytic approach and renamed the subject the calculus of variations in his 1756 lecture Elementa Calculi Variationum.[3][4][b]
Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. Isaac Newton and Gottfried Leibniz also gave some early attention to the subject.[5] To this discrimination Vincenzo Brunacci (1810), Carl Friedrich Gauss (1829), Siméon Poisson (1831), Mikhail Ostrogradsky (1834), and Carl Jacobi (1837) have been among the contributors. An important general work is that of Sarrus (1842) which was condensed and improved by Cauchy (1844). Other valuable treatises and memoirs have been written by Strauch (1849), Jellett (1850), Otto Hesse (1857), Alfred Clebsch (1858), and Lewis Buffett Carll (1885), but perhaps the most important work of the century is that of Weierstrass. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation. The 20th and the 23rdHilbert problem published in 1900 encouraged further development.[5]
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 of a given function space defined over a given domain. A functional is said to have an extremum at the function if has the same sign for all in an arbitrarily small neighborhood of [d] The function is called an extremal function or extremal.[e] The extremum is called a local maximum if everywhere in an arbitrarily small neighborhood of and a local minimum if 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.[11]
Both strong and weak extrema of functionals are for a space of continuous functions but strong extrema have the additional requirement that the first derivatives of the functions in the space be continuous. Thus a strong extremum is also a weak extremum, but the converse may not hold. Finding strong extrema is more difficult than finding weak extrema.[12] An example of a necessary condition that is used for finding weak extrema is the Euler–Lagrange equation.[13][f]
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.[g]
is twice continuously differentiable with respect to its arguments and
If the functional attains a local minimum at and is an arbitrary function that has at least one derivative and vanishes at the endpoints and then for any number close to 0,
The term is called the variation of the function and is denoted by [1][h]
Substituting for in the functional the result is a function of
Since the functional has a minimum for the function has a minimum at and thus,[i]
Taking the total derivative of where and are considered as functions of rather than yields
and because and
Therefore,
where when and we have used integration by parts on the second term. The second term on the second line vanishes because at and by definition. Also, as previously mentioned the left side of the equation is zero so that
In order to illustrate this process, consider the problem of finding the extremal function which is the shortest curve that connects two points and The arc length of the curve is given by
with
Note that assuming y is a function of x 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 that minimizes the functional
with
Since does not appear explicitly in the first term in the Euler–Lagrange equation vanishes for all and thus,
Substituting for and taking the derivative,
Thus
for some constant Then
where
Solving, we get
which implies that
is a constant and therefore that the shortest curve that connects two points and is
and we have thus found the extremal function that minimizes the functional so that is a minimum. The equation for a straight line is In other words, the shortest distance between two points is a straight line.[j]
Beltrami's identityedit
In physics problems it may be the case that meaning the integrand is a function of and but does not appear separately. In that case, the Euler–Lagrange equation can be simplified to the Beltrami identity[16]
The intuition behind this result is that, if the variable is actually time, then the statement implies that the Lagrangian is time-independent. By Noether's theorem, there is an associated conserved quantity. In this case, this quantity is the Hamiltonian, the Legendre transform of the Lagrangian, which (often) coincides with the energy of the system. This is (minus) the constant in Beltrami's identity.
The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integral 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 has continuous first and second derivatives with respect to all of its arguments, and if
then has two continuous derivatives, and it satisfies the Euler–Lagrange equation.
Lavrentiev phenomenonedit
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:[18]
Clearly, minimizes the functional, but we find any function gives a value bounded away from the infimum.
Examples (in one-dimension) are traditionally manifested across and but Ball and Mizel[19] procured the first functional that displayed Lavrentiev's Phenomenon across and for 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.[20]
Functions of several variablesedit
For example, if denotes the displacement of a membrane above the domain in the plane, then its potential energy is proportional to its surface area:
Plateau's problem consists of finding a function that minimizes the surface area while assuming prescribed values on the boundary of ; the solutions are called minimal surfaces. The Euler–Lagrange equation for this problem is nonlinear:
See Courant (1950) for details.
Dirichlet's principleedit
It is often sufficient to consider only small displacements of the membrane, whose energy difference from no displacement is approximated by
The functional is to be minimized among all trial functions that assume prescribed values on the boundary of If is the minimizing function and is an arbitrary smooth function that vanishes on the boundary of then the first variation of must vanish:
Provided that u has two derivatives, we may apply the divergence theorem to obtain
where is the boundary of is arclength along and is the normal derivative of on Since vanishes on and the first variation vanishes, the result is
for all smooth functions v that vanish on the boundary of The proof for the case of one dimensional integrals may be adapted to this case to show that
in
The difficulty with this reasoning is the assumption that the minimizing function u must have two derivatives. Riemann argued that the existence of a smooth minimizing function was assured by the connection with the physical problem: membranes do indeed assume configurations with minimal potential energy. Riemann named this idea the Dirichlet principle in honor of his teacher Peter Gustav Lejeune Dirichlet. However Weierstrass gave an example of a variational problem with no solution: minimize
among all functions that satisfy and 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 [k] Eventually it was shown that Dirichlet's principle is valid, but it requires a sophisticated application of the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998).
Generalization to other boundary value problemsedit
A more general expression for the potential energy of a membrane is
This corresponds to an external force density in an external force on the boundary and elastic forces with modulus acting on The function that minimizes the potential energy with no restriction on its boundary values will be denoted by Provided that and are continuous, regularity theory implies that the minimizing function will have two derivatives. In taking the first variation, no boundary condition need be imposed on the increment The first variation of is given by
If we apply the divergence theorem, the result is
If we first set on the boundary integral vanishes, and we conclude as before that
in Then if we allow to assume arbitrary boundary values, this implies that must satisfy the boundary condition
on This boundary condition is a consequence of the minimizing property of : it is not imposed beforehand. Such conditions are called natural boundary conditions.
The preceding reasoning is not valid if vanishes identically on In such a case, we could allow a trial function where is a constant. For such a trial function,
By appropriate choice of can assume any value unless the quantity inside the brackets vanishes. Therefore, the variational problem is meaningless unless
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 problemsedit
Both one-dimensional and multi-dimensional eigenvalue problems can be formulated as variational problems.
The Sturm–Liouville eigenvalue problem involves a general quadratic form
where is restricted to functions that satisfy the boundary conditions
Let be a normalization integral
The functions and are required to be everywhere positive and bounded away from zero. The primary variational problem is to minimize the ratio among all satisfying the endpoint conditions, which is equivalent to minimizing under the constraint that is constant. It is shown below that the Euler–Lagrange equation for the minimizing is
where is the quotient
It can be shown (see Gelfand and Fomin 1963) that the minimizing has two derivatives and satisfies the Euler–Lagrange equation. The associated will be denoted by ; it is the lowest eigenvalue for this equation and boundary conditions. The associated minimizing function will be denoted by This variational characterization of eigenvalues leads to the Rayleigh–Ritz method: choose an approximating 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 under the additional constraint
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 vanish at the endpoints, we may not impose any condition at the endpoints, and set
where and are arbitrary. If we set the first variation for the ratio is
where λ is given by the ratio as previously. After integration by parts,
If we first require that vanish at the endpoints, the first variation will vanish for all such only if
If satisfies this condition, then the first variation will vanish for arbitrary only if
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 dimensionsedit
Eigenvalue problems in higher dimensions are defined in analogy with the one-dimensional case. For example, given a domain with boundary in three dimensions we may define
and
Let be the function that minimizes the quotient with no condition prescribed on the boundary The Euler–Lagrange equation satisfied by is
where
The minimizing must also satisfy the natural boundary condition
on the boundary 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).
Applicationsedit
Opticsedit
Fermat's principle states that light takes a path that (locally) minimizes the optical length between its endpoints. If the -coordinate is chosen as the parameter along the path, and along the path, then the optical length is given by
where the refractive index depends upon the material. If we try then the first variation of (the derivative of with respect to ε) is
After integration by parts of the first term within brackets, we obtain the Euler–Lagrange equation
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 lawedit
There is a discontinuity of the refractive index when light enters or leaves a lens. Let
where and are constants. Then the Euler–Lagrange equation holds as before in the region where or and in fact the path is a straight line there, since the refractive index is constant. At the must be continuous, but may be discontinuous. After integration by parts in the separate regions and using the Euler–Lagrange equations, the first variation takes the form
The factor multiplying is the sine of angle of the incident ray with the axis, and the factor multiplying is the sine of angle of the refracted ray with the 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 dimensionsedit
It is expedient to use vector notation: let let be a parameter, let be the parametric representation of a curve and let be its tangent vector. The optical length of the curve is given by
Note that this integral is invariant with respect to changes in the parametric representation of The Euler–Lagrange equations for a minimizing curve have the symmetric form
where
It follows from the definition that satisfies
Therefore, the integral may also be written as
This form suggests that if we can find a function whose gradient is given by then the integral is given by the difference of 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 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.
where is the velocity, which generally depends upon Wave fronts for light are characteristic surfaces for this partial differential equation: they satisfy
along a system of curves (the light rays) that are given by
These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification
We conclude that the function is the value of the minimizing integral 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.
In classical mechanics, the action, is defined as the time integral of the Lagrangian, The Lagrangian is the difference of energies,
where is the kinetic energy of a mechanical system and its potential energy. Hamilton's principle (or the action principle) states that the motion of a conservative holonomic (integrable constraints) mechanical system is such that the action integral
is stationary with respect to variations in the path The Euler–Lagrange equations for this system are known as Lagrange's equations:
and they are equivalent to Newton's equations of motion (for such systems).
The conjugate momenta are defined by
For example, if
then
Hamiltonian mechanics results if the conjugate momenta are introduced in place of by a Legendre transformation of the Lagrangian into the Hamiltonian defined by
The Hamiltonian is the total energy of the system: 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 This function is a solution of the Hamilton–Jacobi equation:
Further applicationsedit
Further applications of the calculus of variations include the following:
Variations and sufficient condition for a minimumedit
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[l] is defined as the linear part of the change in the functional, and the second variation[m] is defined as the quadratic part.[22]
For example, if is a functional with the function as its argument, and there is a small change in its argument from to where is a function in the same function space as then the corresponding change in the functional is[n]
The functional is said to be differentiable if
where is a linear functional,[o] is the norm of [p] and as The linear functional is the first variation of and is denoted by,[26]
The functional is said to be twice differentiable if
where is a linear functional (the first variation), is a quadratic functional,[q] and as The quadratic functional is the second variation of and is denoted by,[28]
The second variation is said to be strongly positive if
Using the above definitions, especially the definitions of first variation, second variation, and strongly positive, the following sufficient condition for a minimum of a functional can be stated.
Sufficient condition for a minimum:
The functional has a minimum at if its first variation at and its second variation is strongly positive at [30][r][s]
^Whereas elementary calculus is about infinitesimally small changes in the values of functions without changes in the function itself, calculus of variations is about infinitesimally small changes in the function itself, which are called variations.[1]
^"Euler waited until Lagrange had published on the subject in 1762 ... before he committed his lecture ... to print, so as not to rob Lagrange of his glory. Indeed, it was only Lagrange's method that Euler called Calculus of Variations."[3]
^See Harold J. Kushner (2004): regarding Dynamic Programming, "The calculus of variations had related ideas (e.g., the work of Caratheodory, the Hamilton-Jacobi equation). This led to conflicts with the calculus of variations community."
^The neighborhood of is the part of the given function space where over the whole domain of the functions, with a positive number that specifies the size of the neighborhood.[10]
^ Note the difference between the terms extremal and extremum. An extremal is a function that makes a functional an extremum.
^The following derivation of the Euler–Lagrange equation corresponds to the derivation on pp. 184–185 of Courant & Hilbert (1953).[14]
^Note that and are evaluated at the same values of which is not valid more generally in variational calculus with non-holonomic constraints.
^The product is called the first variation of the functional and is denoted by Some references define the first variation differently by leaving out the factor.
^ As a historical note, this is an axiom of Archimedes. See e.g. Kelland (1843).[15]
^The resulting controversy over the validity of Dirichlet's principle is explained by Turnbull.[21]
^ The first variation is also called the variation, differential, or first differential.
^ The second variation is also called the second differential.
^Note that and the variations below, depend on both and The argument has been left out to simplify the notation. For example, could have been written [23]
^A functional is said to be linear if and where are functions and is a real number.[24]
^ For a function that is defined for where and are real numbers, the norm of is its maximum absolute value, i.e. [25]
^ A functional is said to be quadratic if it is a bilinear functional with two argument functions that are equal. A bilinear functional is a functional that depends on two argument functions and is linear when each argument function in turn is fixed while the other argument function is variable.[27]
Chapter5: "The Second Variation. Sufficient Conditions for a Weak Extremum" – Sufficient conditions for a weak minimum are given by the theorem on p.116.
Chapter6: "Fields. Sufficient Conditions for a Strong Extremum" – Sufficient conditions for a strong minimum are given by the theorem on p.148.
^ One may note the similarity to the sufficient condition for a minimum of a function, where the first derivative is zero and the second derivative is positive.
^ abThiele, Rüdiger (2007). "Euler and the Calculus of Variations". In Bradley, Robert E.; Sandifer, C. Edward (eds.). Leonhard Euler: Life, Work and Legacy. Elsevier. p. 249. ISBN9780080471297.
^Goldstine, Herman H. (2012). A History of the Calculus of Variations from the 17th through the 19th Century. Springer Science & Business Media. p. 110. ISBN9781461381068.
^ abFerguson, James (2004). "Brief Survey of the History of the Calculus of Variations and its Applications". arXiv:math/0402357.
^Dimitri Bertsekas. Dynamic programming and optimal control. Athena Scientific, 2005.
^Bellman, Richard E. (1954). "Dynamic Programming and a new formalism in the calculus of variations". Proc. Natl. Acad. Sci. 40 (4): 231–235. Bibcode:1954PNAS...40..231B. doi:10.1073/pnas.40.4.231. PMC527981. PMID 16589462.
^. American Automatic Control Council. 2004. Archived from the original on 2018-10-01. Retrieved 2013-07-28.
April 11, 2024
calculus, variations, variational, method, redirects, here, approximation, method, quantum, mechanics, variational, method, quantum, mechanics, calculus, variations, variational, calculus, field, mathematical, analysis, that, uses, variations, which, small, ch. Variational method redirects here For the use as an approximation method in quantum mechanics see Variational method quantum mechanics The calculus of variations or variational calculus is a field of mathematical analysis that uses variations which are small changes in functions and functionals to find maxima and minima of functionals mappings from a set of functions to the real numbers a Functionals are often expressed as definite integrals involving functions and their derivatives Functions that maximize or minimize functionals may be found using the Euler Lagrange equation of the calculus of variations A simple example of such a problem is to find the curve of shortest length connecting two points If there are no constraints the solution is a straight line between the points However if the curve is constrained to lie on a surface in space then the solution is less obvious and possibly many solutions may exist Such solutions are known as geodesics A related problem is posed by Fermat s principle light follows the path of shortest optical length connecting two points which depends upon the material of the medium One corresponding concept in mechanics is the principle of least stationary action Many important problems involve functions of several variables Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet s principle Plateau s problem requires finding a surface of minimal area that spans a given contour in space a solution can often be found by dipping a frame in soapy water Although such experiments are relatively easy to perform their mathematical formulation is far from simple there may be more than one locally minimizing surface and they may have non trivial topology Contents 1 History 2 Extrema 3 Euler Lagrange equation 3 1 Example 4 Beltrami s identity 5 Euler Poisson equation 6 Du Bois Reymond s theorem 7 Lavrentiev phenomenon 8 Functions of several variables 8 1 Dirichlet s principle 8 2 Generalization to other boundary value problems 9 Eigenvalue problems 9 1 Sturm Liouville problems 9 2 Eigenvalue problems in several dimensions 10 Applications 10 1 Optics 10 1 1 Snell s law 10 1 2 Fermat s principle in three dimensions 10 1 2 1 Connection with the wave equation 10 2 Mechanics 10 3 Further applications 11 Variations and sufficient condition for a minimum 12 See also 13 Notes 14 References 15 Further reading 16 External linksHistory editThe calculus of variations may be said to begin with Newton s minimal resistance problem in 1687 followed by the brachistochrone curve problem raised by Johann Bernoulli 1696 2 It immediately occupied the attention of Jacob Bernoulli and the Marquis de l Hopital but Leonhard Euler first elaborated the subject beginning in 1733 Lagrange was influenced by Euler s work to contribute significantly to the theory After Euler saw the 1755 work of the 19 year old Lagrange Euler dropped his own partly geometric approach in favor of Lagrange s purely analytic approach and renamed the subject the calculus of variations in his 1756 lecture Elementa Calculi Variationum 3 4 b Legendre 1786 laid down a method not entirely satisfactory for the discrimination of maxima and minima Isaac Newton and Gottfried Leibniz also gave some early attention to the subject 5 To this discrimination Vincenzo Brunacci 1810 Carl Friedrich Gauss 1829 Simeon Poisson 1831 Mikhail Ostrogradsky 1834 and Carl Jacobi 1837 have been among the contributors An important general work is that of Sarrus 1842 which was condensed and improved by Cauchy 1844 Other valuable treatises and memoirs have been written by Strauch 1849 Jellett 1850 Otto Hesse 1857 Alfred Clebsch 1858 and Lewis Buffett Carll 1885 but perhaps the most important work of the century is that of Weierstrass His celebrated course on the theory is epoch making and it may be asserted that he was the first to place it on a firm and unquestionable foundation The 20th and the 23rd Hilbert problem published in 1900 encouraged further development 5 In the 20th century David Hilbert Oskar Bolza Gilbert Ames Bliss Emmy Noether Leonida Tonelli Henri Lebesgue and Jacques Hadamard among others made significant contributions 5 Marston Morse applied calculus of variations in what is now called Morse theory 6 Lev Pontryagin Ralph Rockafellar and F H Clarke developed new mathematical tools for the calculus of variations in optimal control theory 6 The dynamic programming of Richard Bellman is an alternative to the calculus of variations 7 8 9 c Extrema editThe 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 displaystyle y nbsp of a given function space defined over a given domain A functional J y displaystyle J y nbsp is said to have an extremum at the function f displaystyle f nbsp if DJ J y J f displaystyle Delta J J y J f nbsp has the same sign for all y displaystyle y nbsp in an arbitrarily small neighborhood of f displaystyle f nbsp d The function f displaystyle f nbsp is called an extremal function or extremal e The extremum J f displaystyle J f nbsp is called a local maximum if DJ 0 displaystyle Delta J leq 0 nbsp everywhere in an arbitrarily small neighborhood of f displaystyle f nbsp and a local minimum if DJ 0 displaystyle Delta J geq 0 nbsp 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 11 Both strong and weak extrema of functionals are for a space of continuous functions but strong extrema have the additional requirement that the first derivatives of the functions in the space be continuous Thus a strong extremum is also a weak extremum but the converse may not hold Finding strong extrema is more difficult than finding weak extrema 12 An example of a necessary condition that is used for finding weak extrema is the Euler Lagrange equation 13 f Euler Lagrange equation editMain article Euler Lagrange equation Finding the extrema of functionals is similar to finding the maxima and minima of functions The maxima and minima of a function may be located by finding the points where its derivative vanishes i e is equal to zero The extrema of functionals may be obtained by finding functions for which the functional derivative is equal to zero This leads to solving the associated Euler Lagrange equation g Consider the functionalJ y x1x2L x y x y x dx displaystyle J y int x 1 x 2 L left x y x y x right dx nbsp where x1 x2 displaystyle x 1 x 2 nbsp are constants y x displaystyle y x nbsp is twice continuously differentiable y x dydx displaystyle y x frac dy dx nbsp L x y x y x displaystyle L left x y x y x right nbsp is twice continuously differentiable with respect to its arguments x y displaystyle x y nbsp and y displaystyle y nbsp If the functional J y displaystyle J y nbsp attains a local minimum at f displaystyle f nbsp and h x displaystyle eta x nbsp is an arbitrary function that has at least one derivative and vanishes at the endpoints x1 displaystyle x 1 nbsp and x2 displaystyle x 2 nbsp then for any number e displaystyle varepsilon nbsp close to 0 J f J f eh displaystyle J f leq J f varepsilon eta nbsp The term eh displaystyle varepsilon eta nbsp is called the variation of the function f displaystyle f nbsp and is denoted by df displaystyle delta f nbsp 1 h Substituting f eh displaystyle f varepsilon eta nbsp for y displaystyle y nbsp in the functional J y displaystyle J y nbsp the result is a function of e displaystyle varepsilon nbsp F e J f eh displaystyle Phi varepsilon J f varepsilon eta nbsp Since the functional J y displaystyle J y nbsp has a minimum for y f displaystyle y f nbsp the function F e displaystyle Phi varepsilon nbsp has a minimum at e 0 displaystyle varepsilon 0 nbsp and thus i F 0 dFde e 0 x1x2dLde e 0dx 0 displaystyle Phi 0 equiv left frac d Phi d varepsilon right varepsilon 0 int x 1 x 2 left frac dL d varepsilon right varepsilon 0 dx 0 nbsp Taking the total derivative of L x y y displaystyle L left x y y right nbsp where y f eh displaystyle y f varepsilon eta nbsp and y f eh displaystyle y f varepsilon eta nbsp are considered as functions of e displaystyle varepsilon nbsp rather than x displaystyle x nbsp yieldsdLde L ydyde L y dy de displaystyle frac dL d varepsilon frac partial L partial y frac dy d varepsilon frac partial L partial y frac dy d varepsilon nbsp and because dyde h displaystyle frac dy d varepsilon eta nbsp and dy de h displaystyle frac dy d varepsilon eta nbsp dLde L yh L y h displaystyle frac dL d varepsilon frac partial L partial y eta frac partial L partial y eta nbsp Therefore x1x2dLde e 0dx x1x2 L fh L f h dx x1x2 L fhdx L f h x1x2 x1x2hddx L f dx x1x2 L fh hddx L f dx displaystyle begin aligned int x 1 x 2 left frac dL d varepsilon right varepsilon 0 dx amp int x 1 x 2 left frac partial L partial f eta frac partial L partial f eta right dx amp int x 1 x 2 frac partial L partial f eta dx left frac partial L partial f eta right x 1 x 2 int x 1 x 2 eta frac d dx frac partial L partial f dx amp int x 1 x 2 left frac partial L partial f eta eta frac d dx frac partial L partial f right dx end aligned nbsp where L x y y L x f f displaystyle L left x y y right to L left x f f right nbsp when e 0 displaystyle varepsilon 0 nbsp and we have used integration by parts on the second term The second term on the second line vanishes because h 0 displaystyle eta 0 nbsp at x1 displaystyle x 1 nbsp and x2 displaystyle x 2 nbsp by definition Also as previously mentioned the left side of the equation is zero so that x1x2h x L f ddx L f dx 0 displaystyle int x 1 x 2 eta x left frac partial L partial f frac d dx frac partial L partial f right dx 0 nbsp According to the fundamental lemma of calculus of variations the part of the integrand in parentheses is zero i e L f ddx L f 0 displaystyle frac partial L partial f frac d dx frac partial L partial f 0 nbsp which is called the Euler Lagrange equation The left hand side of this equation is called the functional derivative of J f displaystyle J f nbsp and is denoted dJ df x displaystyle delta J delta f x nbsp In general this gives a second order ordinary differential equation which can be solved to obtain the extremal function f x displaystyle f x nbsp The Euler Lagrange equation is a necessary but not sufficient condition for an extremum J f displaystyle J f nbsp A sufficient condition for a minimum is given in the section Variations and sufficient condition for a minimum Example edit In order to illustrate this process consider the problem of finding the extremal function y f x displaystyle y f x nbsp which is the shortest curve that connects two points x1 y1 displaystyle left x 1 y 1 right nbsp and x2 y2 displaystyle left x 2 y 2 right nbsp The arc length of the curve is given byA y x1x21 y x 2dx displaystyle A y int x 1 x 2 sqrt 1 y x 2 dx nbsp with y x dydx y1 f x1 y2 f x2 displaystyle y x frac dy dx y 1 f x 1 y 2 f x 2 nbsp Note that assuming y is a function of x 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 displaystyle f x nbsp that minimizes the functional A y displaystyle A y nbsp L f ddx L f 0 displaystyle frac partial L partial f frac d dx frac partial L partial f 0 nbsp with L 1 f x 2 displaystyle L sqrt 1 f x 2 nbsp Since f displaystyle f nbsp does not appear explicitly in L displaystyle L nbsp the first term in the Euler Lagrange equation vanishes for all f x displaystyle f x nbsp and thus ddx L f 0 displaystyle frac d dx frac partial L partial f 0 nbsp Substituting for L displaystyle L nbsp and taking the derivative ddx f x 1 f x 2 0 displaystyle frac d dx frac f x sqrt 1 f x 2 0 nbsp Thusf x 1 f x 2 c displaystyle frac f x sqrt 1 f x 2 c nbsp for some constant c displaystyle c nbsp Then f x 21 f x 2 c2 displaystyle frac f x 2 1 f x 2 c 2 nbsp where 0 c2 lt 1 displaystyle 0 leq c 2 lt 1 nbsp Solving we get f x 2 c21 c2 displaystyle f x 2 frac c 2 1 c 2 nbsp which implies that f x m displaystyle f x m nbsp is a constant and therefore that the shortest curve that connects two points x1 y1 displaystyle left x 1 y 1 right nbsp and x2 y2 displaystyle left x 2 y 2 right nbsp is f x mx bwith m y2 y1x2 x1andb x2y1 x1y2x2 x1 displaystyle f x mx b qquad text with m frac y 2 y 1 x 2 x 1 quad text and quad b frac x 2 y 1 x 1 y 2 x 2 x 1 nbsp and we have thus found the extremal function f x displaystyle f x nbsp that minimizes the functional A y displaystyle A y nbsp so that A f displaystyle A f nbsp is a minimum The equation for a straight line is y f x displaystyle y f x nbsp In other words the shortest distance between two points is a straight line j Beltrami s identity editIn physics problems it may be the case that L x 0 displaystyle frac partial L partial x 0 nbsp meaning the integrand is a function of f x displaystyle f x nbsp and f x displaystyle f x nbsp but x displaystyle x nbsp does not appear separately In that case the Euler Lagrange equation can be simplified to the Beltrami identity 16 L f L f C displaystyle L f frac partial L partial f C nbsp where C displaystyle C nbsp is a constant The left hand side is the Legendre transformation of L displaystyle L nbsp with respect to f x displaystyle f x nbsp The intuition behind this result is that if the variable x displaystyle x nbsp is actually time then the statement L x 0 displaystyle frac partial L partial x 0 nbsp implies that the Lagrangian is time independent By Noether s theorem there is an associated conserved quantity In this case this quantity is the Hamiltonian the Legendre transform of the Lagrangian which often coincides with the energy of the system This is minus the constant in Beltrami s identity Euler Poisson equation editIf S displaystyle S nbsp depends on higher derivatives of y x displaystyle y x nbsp that is ifS abf x y x y x y n x dx displaystyle S int a b f x y x y x dots y n x dx nbsp then y displaystyle y nbsp must satisfy the Euler Poisson equation 17 f y ddx f y 1 ndndxn f y n 0 displaystyle frac partial f partial y frac d dx left frac partial f partial y right dots 1 n frac d n dx n left frac partial f partial y n right 0 nbsp Du Bois Reymond s theorem editThe discussion thus far has assumed that extremal functions possess two continuous derivatives although the existence of the integral J displaystyle J nbsp 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 displaystyle L nbsp has continuous first and second derivatives with respect to all of its arguments and if 2L f 2 0 displaystyle frac partial 2 L partial f 2 neq 0 nbsp then f displaystyle f nbsp has two continuous derivatives and it satisfies the Euler Lagrange equation Lavrentiev phenomenon editHilbert 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 Mania in 1934 18 L x 01 x3 t 2x 6 displaystyle L x int 0 1 x 3 t 2 x 6 nbsp A x W1 1 0 1 x 0 0 x 1 1 displaystyle A x in W 1 1 0 1 x 0 0 x 1 1 nbsp Clearly x t t13 displaystyle x t t frac 1 3 nbsp minimizes the functional but we find any function x W1 displaystyle x in W 1 infty nbsp gives a value bounded away from the infimum Examples in one dimension are traditionally manifested across W1 1 displaystyle W 1 1 nbsp and W1 displaystyle W 1 infty nbsp but Ball and Mizel 19 procured the first functional that displayed Lavrentiev s Phenomenon across W1 p displaystyle W 1 p nbsp and W1 q displaystyle W 1 q nbsp for 1 p lt q lt displaystyle 1 leq p lt q lt infty nbsp 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 20 Functions of several variables editFor example if f x y displaystyle varphi x y nbsp denotes the displacement of a membrane above the domain D displaystyle D nbsp in the x y displaystyle x y nbsp plane then its potential energy is proportional to its surface area U f D1 f fdxdy displaystyle U varphi iint D sqrt 1 nabla varphi cdot nabla varphi dx dy nbsp Plateau s problem consists of finding a function that minimizes the surface area while assuming prescribed values on the boundary of D displaystyle D nbsp the solutions are called minimal surfaces The Euler Lagrange equation for this problem is nonlinear fxx 1 fy2 fyy 1 fx2 2fxfyfxy 0 displaystyle varphi xx 1 varphi y 2 varphi yy 1 varphi x 2 2 varphi x varphi y varphi xy 0 nbsp See Courant 1950 for details Dirichlet s principle edit It is often sufficient to consider only small displacements of the membrane whose energy difference from no displacement is approximated byV f 12 D f fdxdy displaystyle V varphi frac 1 2 iint D nabla varphi cdot nabla varphi dx dy nbsp The functional V displaystyle V nbsp is to be minimized among all trial functions f displaystyle varphi nbsp that assume prescribed values on the boundary of D displaystyle D nbsp If u displaystyle u nbsp is the minimizing function and v displaystyle v nbsp is an arbitrary smooth function that vanishes on the boundary of D displaystyle D nbsp then the first variation of V u ev displaystyle V u varepsilon v nbsp must vanish ddeV u ev e 0 D u vdxdy 0 displaystyle left frac d d varepsilon V u varepsilon v right varepsilon 0 iint D nabla u cdot nabla v dx dy 0 nbsp Provided that u has two derivatives we may apply the divergence theorem to obtain D v u dxdy D u v v udxdy Cv u nds displaystyle iint D nabla cdot v nabla u dx dy iint D nabla u cdot nabla v v nabla cdot nabla u dx dy int C v frac partial u partial n ds nbsp where C displaystyle C nbsp is the boundary of D displaystyle D nbsp s displaystyle s nbsp is arclength along C displaystyle C nbsp and u n displaystyle partial u partial n nbsp is the normal derivative of u displaystyle u nbsp on C displaystyle C nbsp Since v displaystyle v nbsp vanishes on C displaystyle C nbsp and the first variation vanishes the result is Dv udxdy 0 displaystyle iint D v nabla cdot nabla u dx dy 0 nbsp for all smooth functions v that vanish on the boundary of D displaystyle D nbsp The proof for the case of one dimensional integrals may be adapted to this case to show that u 0 displaystyle nabla cdot nabla u 0 nbsp in D displaystyle D nbsp The difficulty with this reasoning is the assumption that the minimizing function u must have two derivatives Riemann argued that the existence of a smooth minimizing function was assured by the connection with the physical problem membranes do indeed assume configurations with minimal potential energy Riemann named this idea the Dirichlet principle in honor of his teacher Peter Gustav Lejeune Dirichlet However Weierstrass gave an example of a variational problem with no solution minimizeW f 11 xf 2dx displaystyle W varphi int 1 1 x varphi 2 dx nbsp among all functions f displaystyle varphi nbsp that satisfy f 1 1 displaystyle varphi 1 1 nbsp and f 1 1 displaystyle varphi 1 1 nbsp W displaystyle W nbsp 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 displaystyle W 0 nbsp k Eventually it was shown that Dirichlet s principle is valid but it requires a sophisticated application of the regularity theory for elliptic partial differential equations see Jost and Li Jost 1998 Generalization to other boundary value problems edit A more general expression for the potential energy of a membrane isV f D 12 f f f x y f dxdy C 12s s f2 g s f ds displaystyle V varphi iint D left frac 1 2 nabla varphi cdot nabla varphi f x y varphi right dx dy int C left frac 1 2 sigma s varphi 2 g s varphi right ds nbsp This corresponds to an external force density f x y displaystyle f x y nbsp in D displaystyle D nbsp an external force g s displaystyle g s nbsp on the boundary C displaystyle C nbsp and elastic forces with modulus s s displaystyle sigma s nbsp acting on C displaystyle C nbsp The function that minimizes the potential energy with no restriction on its boundary values will be denoted by u displaystyle u nbsp Provided that f displaystyle f nbsp and g displaystyle g nbsp are continuous regularity theory implies that the minimizing function u displaystyle u nbsp will have two derivatives In taking the first variation no boundary condition need be imposed on the increment v displaystyle v nbsp The first variation of V u ev displaystyle V u varepsilon v nbsp is given by D u v fv dxdy C suv gv ds 0 displaystyle iint D left nabla u cdot nabla v fv right dx dy int C left sigma uv gv right ds 0 nbsp If we apply the divergence theorem the result is D v u vf dxdy Cv u n su g ds 0 displaystyle iint D left v nabla cdot nabla u vf right dx dy int C v left frac partial u partial n sigma u g right ds 0 nbsp If we first set v 0 displaystyle v 0 nbsp on C displaystyle C nbsp the boundary integral vanishes and we conclude as before that u f 0 displaystyle nabla cdot nabla u f 0 nbsp in D displaystyle D nbsp Then if we allow v displaystyle v nbsp to assume arbitrary boundary values this implies that u displaystyle u nbsp must satisfy the boundary condition u n su g 0 displaystyle frac partial u partial n sigma u g 0 nbsp on C displaystyle C nbsp This boundary condition is a consequence of the minimizing property of u displaystyle u nbsp it is not imposed beforehand Such conditions are called natural boundary conditions The preceding reasoning is not valid if s displaystyle sigma nbsp vanishes identically on C displaystyle C nbsp In such a case we could allow a trial function f c displaystyle varphi equiv c nbsp where c displaystyle c nbsp is a constant For such a trial function V c c Dfdxdy Cgds displaystyle V c c left iint D f dx dy int C g ds right nbsp By appropriate choice of c displaystyle c nbsp V displaystyle V nbsp can assume any value unless the quantity inside the brackets vanishes Therefore the variational problem is meaningless unless Dfdxdy Cgds 0 displaystyle iint D f dx dy int C g ds 0 nbsp 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 editBoth one dimensional and multi dimensional eigenvalue problems can be formulated as variational problems Sturm Liouville problems edit See also Sturm Liouville theory The Sturm Liouville eigenvalue problem involves a general quadratic formQ y x1x2 p x y x 2 q x y x 2 dx displaystyle Q y int x 1 x 2 left p x y x 2 q x y x 2 right dx nbsp where y displaystyle y nbsp is restricted to functions that satisfy the boundary conditions y x1 0 y x2 0 displaystyle y x 1 0 quad y x 2 0 nbsp Let R displaystyle R nbsp be a normalization integral R y x1x2r x y x 2dx displaystyle R y int x 1 x 2 r x y x 2 dx nbsp The functions p x displaystyle p x nbsp and r x displaystyle r x nbsp are required to be everywhere positive and bounded away from zero The primary variational problem is to minimize the ratio Q R displaystyle Q R nbsp among all y displaystyle y nbsp satisfying the endpoint conditions which is equivalent to minimizing Q y displaystyle Q y nbsp under the constraint that R y displaystyle R y nbsp is constant It is shown below that the Euler Lagrange equation for the minimizing u displaystyle u nbsp is pu qu lru 0 displaystyle pu qu lambda ru 0 nbsp where l displaystyle lambda nbsp is the quotient l Q u R u displaystyle lambda frac Q u R u nbsp It can be shown see Gelfand and Fomin 1963 that the minimizing u displaystyle u nbsp has two derivatives and satisfies the Euler Lagrange equation The associated l displaystyle lambda nbsp will be denoted by l1 displaystyle lambda 1 nbsp it is the lowest eigenvalue for this equation and boundary conditions The associated minimizing function will be denoted by u1 x displaystyle u 1 x nbsp This variational characterization of eigenvalues leads to the Rayleigh Ritz method choose an approximating u displaystyle u nbsp 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 displaystyle Q nbsp under the additional constraint x1x2r x u1 x y x dx 0 displaystyle int x 1 x 2 r x u 1 x y x dx 0 nbsp 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 displaystyle y nbsp vanish at the endpoints we may not impose any condition at the endpoints and setQ y x1x2 p x y x 2 q x y x 2 dx a1y x1 2 a2y x2 2 displaystyle Q y int x 1 x 2 left p x y x 2 q x y x 2 right dx a 1 y x 1 2 a 2 y x 2 2 nbsp where a1 displaystyle a 1 nbsp and a2 displaystyle a 2 nbsp are arbitrary If we set y u ev displaystyle y u varepsilon v nbsp the first variation for the ratio Q R displaystyle Q R nbsp is V1 2R u x1x2 p x u x v x q x u x v x lr x u x v x dx a1u x1 v x1 a2u x2 v x2 displaystyle V 1 frac 2 R u left int x 1 x 2 left p x u x v x q x u x v x lambda r x u x v x right dx a 1 u x 1 v x 1 a 2 u x 2 v x 2 right nbsp where l is given by the ratio Q u R u displaystyle Q u R u nbsp as previously After integration by parts R u 2V1 x1x2v x pu qu lru dx v x1 p x1 u x1 a1u x1 v x2 p x2 u x2 a2u x2 displaystyle frac R u 2 V 1 int x 1 x 2 v x left pu qu lambda ru right dx v x 1 p x 1 u x 1 a 1 u x 1 v x 2 p x 2 u x 2 a 2 u x 2 nbsp If we first require that v displaystyle v nbsp vanish at the endpoints the first variation will vanish for all such v displaystyle v nbsp only if pu qu lru 0forx1 lt x lt x2 displaystyle pu qu lambda ru 0 quad hbox for quad x 1 lt x lt x 2 nbsp If u displaystyle u nbsp satisfies this condition then the first variation will vanish for arbitrary v displaystyle v nbsp only if p x1 u x1 a1u x1 0 andp x2 u x2 a2u x2 0 displaystyle p x 1 u x 1 a 1 u x 1 0 quad hbox and quad p x 2 u x 2 a 2 u x 2 0 nbsp 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 edit Eigenvalue problems in higher dimensions are defined in analogy with the one dimensional case For example given a domain D displaystyle D nbsp with boundary B displaystyle B nbsp in three dimensions we may defineQ f Dp X f f q X f2dxdydz Bs S f2dS displaystyle 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 nbsp and R f Dr X f X 2dxdydz displaystyle R varphi iiint D r X varphi X 2 dx dy dz nbsp Let u displaystyle u nbsp be the function that minimizes the quotient Q f R f displaystyle Q varphi R varphi nbsp with no condition prescribed on the boundary B displaystyle B nbsp The Euler Lagrange equation satisfied by u displaystyle u nbsp is p X u q x u lr x u 0 displaystyle nabla cdot p X nabla u q x u lambda r x u 0 nbsp where l Q u R u displaystyle lambda frac Q u R u nbsp The minimizing u displaystyle u nbsp must also satisfy the natural boundary condition p S u n s S u 0 displaystyle p S frac partial u partial n sigma S u 0 nbsp on the boundary B displaystyle B nbsp 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 editOptics edit Fermat s principle states that light takes a path that locally minimizes the optical length between its endpoints If the x displaystyle x nbsp coordinate is chosen as the parameter along the path and y f x displaystyle y f x nbsp along the path then the optical length is given byA f x0x1n x f x 1 f x 2dx displaystyle A f int x 0 x 1 n x f x sqrt 1 f x 2 dx nbsp where the refractive index n x y displaystyle n x y nbsp depends upon the material If we try f x f0 x ef1 x displaystyle f x f 0 x varepsilon f 1 x nbsp then the first variation of A displaystyle A nbsp the derivative of A displaystyle A nbsp with respect to e is dA f0 f1 x0x1 n x f0 f0 x f1 x 1 f0 x 2 ny x f0 f11 f0 x 2 dx displaystyle delta A f 0 f 1 int x 0 x 1 left frac n x f 0 f 0 x f 1 x sqrt 1 f 0 x 2 n y x f 0 f 1 sqrt 1 f 0 x 2 right dx nbsp After integration by parts of the first term within brackets we obtain the Euler Lagrange equation ddx n x f0 f0 1 f0 2 ny x f0 1 f0 x 2 0 displaystyle frac d dx left frac n x f 0 f 0 sqrt 1 f 0 2 right n y x f 0 sqrt 1 f 0 x 2 0 nbsp 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 edit There is a discontinuity of the refractive index when light enters or leaves a lens Letn x y n ifx lt 0 n ifx gt 0 displaystyle n x y begin cases n amp text if quad x lt 0 n amp text if quad x gt 0 end cases nbsp where n displaystyle n nbsp and n displaystyle n nbsp are constants Then the Euler Lagrange equation holds as before in the region where x lt 0 displaystyle x lt 0 nbsp or x gt 0 displaystyle x gt 0 nbsp and in fact the path is a straight line there since the refractive index is constant At the x 0 displaystyle x 0 nbsp f displaystyle f nbsp must be continuous but f displaystyle f nbsp may be discontinuous After integration by parts in the separate regions and using the Euler Lagrange equations the first variation takes the form dA f0 f1 f1 0 n f0 0 1 f0 0 2 n f0 0 1 f0 0 2 displaystyle delta A f 0 f 1 f 1 0 left n frac f 0 0 sqrt 1 f 0 0 2 n frac f 0 0 sqrt 1 f 0 0 2 right nbsp The factor multiplying n displaystyle n nbsp is the sine of angle of the incident ray with the x displaystyle x nbsp axis and the factor multiplying n displaystyle n nbsp is the sine of angle of the refracted ray with the x displaystyle x nbsp 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 edit It is expedient to use vector notation let X x1 x2 x3 displaystyle X x 1 x 2 x 3 nbsp let t displaystyle t nbsp be a parameter let X t displaystyle X t nbsp be the parametric representation of a curve C displaystyle C nbsp and let X t displaystyle dot X t nbsp be its tangent vector The optical length of the curve is given byA C t0t1n X X X dt displaystyle A C int t 0 t 1 n X sqrt dot X cdot dot X dt nbsp Note that this integral is invariant with respect to changes in the parametric representation of C displaystyle C nbsp The Euler Lagrange equations for a minimizing curve have the symmetric formddtP X X n displaystyle frac d dt P sqrt dot X cdot dot X nabla n nbsp where P n X X X X displaystyle P frac n X dot X sqrt dot X cdot dot X nbsp It follows from the definition that P displaystyle P nbsp satisfiesP P n X 2 displaystyle P cdot P n X 2 nbsp Therefore the integral may also be written asA C t0t1P X dt displaystyle A C int t 0 t 1 P cdot dot X dt nbsp This form suggests that if we can find a function ps displaystyle psi nbsp whose gradient is given by P displaystyle P nbsp then the integral A displaystyle A nbsp is given by the difference of ps displaystyle psi nbsp 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 ps displaystyle psi nbsp 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 edit The wave equation for an inhomogeneous medium isutt c2 u displaystyle u tt c 2 nabla cdot nabla u nbsp where c displaystyle c nbsp is the velocity which generally depends upon X displaystyle X nbsp Wave fronts for light are characteristic surfaces for this partial differential equation they satisfy ft2 c X 2 f f displaystyle varphi t 2 c X 2 nabla varphi cdot nabla varphi nbsp We may look for solutions in the formf t X t ps X displaystyle varphi t X t psi X nbsp In that case ps displaystyle psi nbsp satisfies ps ps n2 displaystyle nabla psi cdot nabla psi n 2 nbsp where n 1 c displaystyle n 1 c nbsp According to the theory of first order partial differential equations if P ps displaystyle P nabla psi nbsp then P displaystyle P nbsp satisfies dPds n n displaystyle frac dP ds n nabla n nbsp along a system of curves the light rays that are given by dXds P displaystyle frac dX ds P nbsp These equations for solution of a first order partial differential equation are identical to the Euler Lagrange equations if we make the identificationdsdt X X n displaystyle frac ds dt frac sqrt dot X cdot dot X n nbsp We conclude that the function ps displaystyle psi nbsp is the value of the minimizing integral A displaystyle A nbsp 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 edit Main article Action physics In classical mechanics the action S displaystyle S nbsp is defined as the time integral of the Lagrangian L displaystyle L nbsp The Lagrangian is the difference of energies L T U displaystyle L T U nbsp where T displaystyle T nbsp is the kinetic energy of a mechanical system and U displaystyle U nbsp its potential energy Hamilton s principle or the action principle states that the motion of a conservative holonomic integrable constraints mechanical system is such that the action integral S t0t1L x x t dt displaystyle S int t 0 t 1 L x dot x t dt nbsp is stationary with respect to variations in the path x t displaystyle x t nbsp The Euler Lagrange equations for this system are known as Lagrange s equations ddt L x L x displaystyle frac d dt frac partial L partial dot x frac partial L partial x nbsp and they are equivalent to Newton s equations of motion for such systems The conjugate momenta P displaystyle P nbsp are defined byp L x displaystyle p frac partial L partial dot x nbsp For example if T 12mx 2 displaystyle T frac 1 2 m dot x 2 nbsp then p mx displaystyle p m dot x nbsp Hamiltonian mechanics results if the conjugate momenta are introduced in place of x displaystyle dot x nbsp by a Legendre transformation of the Lagrangian L displaystyle L nbsp into the Hamiltonian H displaystyle H nbsp defined by H x p t px L x x t displaystyle H x p t p dot x L x dot x t nbsp The Hamiltonian is the total energy of the system H T U displaystyle H T U nbsp 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 displaystyle X nbsp This function is a solution of the Hamilton Jacobi equation ps t H x ps x t 0 displaystyle frac partial psi partial t H left x frac partial psi partial x t right 0 nbsp Further applications edit Further applications of the calculus of variations include the following The derivation of the catenary shape Solution to Newton s minimal resistance problem Solution to the brachistochrone problem Solution to the tautochrone problem Solution to isoperimetric problems Calculating geodesics Finding minimal surfaces and solving Plateau s problem Optimal control Analytical mechanics or reformulations of Newton s laws of motion most notably Lagrangian and Hamiltonian mechanics Geometric optics especially Lagrangian and Hamiltonian optics Variational method quantum mechanics one way of finding approximations to the lowest energy eigenstate or ground state and some excited states Variational Bayesian methods a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning Variational methods in general relativity a family of techniques using calculus of variations to solve problems in Einstein s general theory of relativity Finite element method is a variational method for finding numerical solutions to boundary value problems in differential equations Total variation denoising an image processing method for filtering high variance or noisy signals Variations and sufficient condition for a minimum editCalculus 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 l is defined as the linear part of the change in the functional and the second variation m is defined as the quadratic part 22 For example if J y displaystyle J y nbsp is a functional with the function y y x displaystyle y y x nbsp as its argument and there is a small change in its argument from y displaystyle y nbsp to y h displaystyle y h nbsp where h h x displaystyle h h x nbsp is a function in the same function space as y displaystyle y nbsp then the corresponding change in the functional is n DJ h J y h J y displaystyle Delta J h J y h J y nbsp The functional J y displaystyle J y nbsp is said to be differentiable ifDJ h f h e h displaystyle Delta J h varphi h varepsilon h nbsp where f h displaystyle varphi h nbsp is a linear functional o h displaystyle h nbsp is the norm of h displaystyle h nbsp p and e 0 displaystyle varepsilon to 0 nbsp as h 0 displaystyle h to 0 nbsp The linear functional f h displaystyle varphi h nbsp is the first variation of J y displaystyle J y nbsp and is denoted by 26 dJ h f h displaystyle delta J h varphi h nbsp The functional J y displaystyle J y nbsp is said to be twice differentiable ifDJ h f1 h f2 h e h 2 displaystyle Delta J h varphi 1 h varphi 2 h varepsilon h 2 nbsp where f1 h displaystyle varphi 1 h nbsp is a linear functional the first variation f2 h displaystyle varphi 2 h nbsp is a quadratic functional q and e 0 displaystyle varepsilon to 0 nbsp as h 0 displaystyle h to 0 nbsp The quadratic functional f2 h displaystyle varphi 2 h nbsp is the second variation of J y displaystyle J y nbsp and is denoted by 28 d2J h f2 h displaystyle delta 2 J h varphi 2 h nbsp The second variation d2J h displaystyle delta 2 J h nbsp is said to be strongly positive ifd2J h k h 2 displaystyle delta 2 J h geq k h 2 nbsp for all h displaystyle h nbsp and for some constant k gt 0 displaystyle k gt 0 nbsp 29 Using the above definitions especially the definitions of first variation second variation and strongly positive the following sufficient condition for a minimum of a functional can be stated Sufficient condition for a minimum The functional J y displaystyle J y nbsp has a minimum at y y displaystyle y hat y nbsp if its first variation dJ h 0 displaystyle delta J h 0 nbsp at y y displaystyle y hat y nbsp and its second variation d2J h displaystyle delta 2 J h nbsp is strongly positive at y y displaystyle y hat y nbsp 30 r s See also editFirst variation Isoperimetric inequality Variational principle Variational bicomplex Fermat s principle Principle of least action Infinite dimensional optimization Finite element method Functional analysis Ekeland s variational principle Inverse problem for Lagrangian mechanics Obstacle problem Perturbation methods Young measure Optimal control Direct method in calculus of variations Noether s theorem De Donder Weyl theory Variational Bayesian methods Chaplygin problem Nehari manifold Hu Washizu principle Luke s variational principle Mountain pass theorem Category Variational analysts Measures of central tendency as solutions to variational problems Stampacchia Medal Fermat Prize Convenient vector spaceNotes edit Whereas elementary calculus is about infinitesimally small changes in the values of functions without changes in the function itself calculus of variations is about infinitesimally small changes in the function itself which are called variations 1 Euler waited until Lagrange had published on the subject in 1762 before he committed his lecture to print so as not to rob Lagrange of his glory Indeed it was only Lagrange s method that Euler called Calculus of Variations 3 See Harold J Kushner 2004 regarding Dynamic Programming The calculus of variations had related ideas e g the work of Caratheodory the Hamilton Jacobi equation This led to conflicts with the calculus of variations community The neighborhood of f displaystyle f nbsp is the part of the given function space where y f lt h displaystyle y f lt h nbsp over the whole domain of the functions with h displaystyle h nbsp a positive number that specifies the size of the neighborhood 10 Note the difference between the terms extremal and extremum An extremal is a function that makes a functional an extremum For a sufficient condition see section Variations and sufficient condition for a minimum The following derivation of the Euler Lagrange equation corresponds to the derivation on pp 184 185 of Courant amp Hilbert 1953 14 Note that h x displaystyle eta x nbsp and f x displaystyle f x nbsp are evaluated at the same values of x displaystyle x nbsp which is not valid more generally in variational calculus with non holonomic constraints The product eF 0 displaystyle varepsilon Phi 0 nbsp is called the first variation of the functional J displaystyle J nbsp and is denoted by dJ displaystyle delta J nbsp Some references define the first variation differently by leaving out the e displaystyle varepsilon nbsp factor As a historical note this is an axiom of Archimedes See e g Kelland 1843 15 The resulting controversy over the validity of Dirichlet s principle is explained by Turnbull 21 The first variation is also called the variation differential or first differential The second variation is also called the second differential Note that DJ h displaystyle Delta J h nbsp and the variations below depend on both y displaystyle y nbsp and h displaystyle h nbsp The argument y displaystyle y nbsp has been left out to simplify the notation For example DJ h displaystyle Delta J h nbsp could have been written DJ y h displaystyle Delta J y h nbsp 23 A functional f h displaystyle varphi h nbsp is said to be linear if f ah af h displaystyle varphi alpha h alpha varphi h nbsp and f h h2 f h f h2 displaystyle varphi left h h 2 right varphi h varphi left h 2 right nbsp where h h2 displaystyle h h 2 nbsp are functions and a displaystyle alpha nbsp is a real number 24 For a function h h x displaystyle h h x nbsp that is defined for a x b displaystyle a leq x leq b nbsp where a displaystyle a nbsp and b displaystyle b nbsp are real numbers the norm of h displaystyle h nbsp is its maximum absolute value i e h maxa x b h x displaystyle h displaystyle max a leq x leq b h x nbsp 25 A functional is said to be quadratic if it is a bilinear functional with two argument functions that are equal A bilinear functional is a functional that depends on two argument functions and is linear when each argument function in turn is fixed while the other argument function is variable 27 For other sufficient conditions see in Gelfand amp Fomin 2000 Chapter 5 The Second Variation Sufficient Conditions for a Weak Extremum Sufficient conditions for a weak minimum are given by the theorem on p 116 Chapter 6 Fields Sufficient Conditions for a Strong Extremum Sufficient conditions for a strong minimum are given by the theorem on p 148 One may note the similarity to the sufficient condition for a minimum of a function where the first derivative is zero and the second derivative is positive References edit a b Courant amp Hilbert 1953 p 184 Gelfand I M Fomin S V 2000 Silverman Richard A ed Calculus of variations Unabridged repr ed Mineola New York Dover Publications p 3 ISBN 978 0486414485 a b Thiele Rudiger 2007 Euler and the Calculus of Variations In Bradley Robert E Sandifer C Edward eds Leonhard Euler Life Work and Legacy Elsevier p 249 ISBN 9780080471297 Goldstine Herman H 2012 A History of the Calculus of Variations from the 17th through the 19th Century Springer Science amp Business Media p 110 ISBN 9781461381068 a b c van Brunt Bruce 2004 The Calculus of Variations Springer ISBN 978 0 387 40247 5 a b Ferguson James 2004 Brief Survey of the History of the Calculus of Variations and its Applications arXiv math 0402357 Dimitri Bertsekas Dynamic programming and optimal control Athena Scientific 2005 Bellman Richard E 1954 Dynamic Programming and a new formalism in the calculus of variations Proc Natl Acad Sci 40 4 231 235 Bibcode 1954PNAS 40 231B doi 10 1073 pnas 40 4 231 PMC 527981 PMID 16589462 Richard E Bellman Control Heritage Award American Automatic Control Council 2004 Archived from the original on 2018 10 01 Retrieved 2013 07 28 li, wikipedia, wiki, book, books, library,