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calculus of variations
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{{redirect|Variational method|the use as an approximation method in quantum mechanics|Variational method (quantum mechanics)}}{{Calculus |specialized}}Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functionsand functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.{{refn|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.{{harvnb|Courant|Hilbert|1953|p=184}}|group="Note"}} 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, where the optical length 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 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 a solution of soap suds. Although such experiments are relatively easy to perform, their mathematical interpretation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.

History

The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696).BOOK, Gelfand, I. M., Israel Gelfand, Fomin, S. V., Sergei Fomin, harv, Calculus of variations, 2000, Dover Publications, Mineola, New York, 978-0486414485, 3,weblink Unabridged repr., Silverman, Richard A., It immediately occupied the attention of Jakob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733. Lagrange was influenced by Euler's work to contribute significantly to the theory. After Euler saw the 1755 work of the 19-year-old Lagrange, Euler dropped his own partly geometric approach in favor of Lagrange's purely analytic approach and renamed the subject the calculus of variations in his 1756 lecture Elementa Calculi Variationum.BOOK, Thiele, Rüdiger, Bradley, Robert E., Sandifer, C. Edward, Leonhard Euler: Life, Work and Legacy, Elsevier, 2007, 249, Euler and the Calculus of Variations,weblink 9780080471297, BOOK, Goldstine, Herman H., 2012, A History of the Calculus of Variations from the 17th through the 19th Century,weblink's%20work%2C%20Euler%20dropped%20his%20own%20method%2C%20espoused%20that%20of%20Lagrange%2C%20and%20renamed%20the%20subject%20the%20calculus%20of%20variations.%22&f=false, Springer Science & Business Media, 110, 9781461381068, Herman Goldstine, {{refn|"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." |group="Note"}} 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.BOOK, van Brunt, Bruce, The Calculus of Variations, Springer, 2004, 978-0-387-40247-5, 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 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. In the 20th century David Hilbert, Emmy Noether, Leonida Tonelli, Henri Lebesgue and Jacques Hadamard among others made significant contributions. Marston Morse applied calculus of variations in what is now called Morse theory.ARXIV, Ferguson, James, math/0402357, Brief Survey of the History of the Calculus of Variations and its Applications, 2004, Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory. The dynamic programming of Richard Bellman is an alternative to the calculus of variations.Dimitri Bertsekas. Dynamic programming and optimal control. Athena Scientific, 2005.JOURNAL, Bellman, Richard E., Dynamic Programming and a new formalism in the calculus of variations, 1954, Proc. Natl. Acad. Sci., 4, 231–235, 527981, 16589462, 40, 10.1073/pnas.40.4.231, 1954PNAS...40..231B, NEWS, Harold J., Kushner, Harold J. Kushner, Richard E. Bellman Control Heritage Award, 2004,weblink American Automatic Control Council, 2013-07-28, See 2004: Harold J. Kushner: 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."

Extrema

The calculus of variations is concerned with the maxima or minima (collectively called extrema) of functionals. A functional maps functions to scalars, so functionals have been described as "functions of functions." Functionals have extrema with respect to the elements {{math|y}} of a given function space defined over a given domain. A functional {{math|J [ y ]}} is said to have an extremum at the function {{math|f  }} if {{math|ΔJ {{=}} J [ y ] − J [ f]}} has the same sign for all {{math|y}} in an arbitrarily small neighborhood of {{math|f .}}{{refn|The neighborhood of {{math|f}} is the part of the given function space where {{math|| y − f| < h}} over the whole domain of the functions, with {{math|h}} a positive number that specifies the size of the neighborhood.BOOK, Courant, R, Richard Courant, Hilbert, D, David Hilbert, Methods of Mathematical Physics, Vol. I, First English, harv, Interscience Publishers, Inc., 1953, New York, 169, 978-0471504474, |group="Note"}} The function {{math|f}} is called an extremal function or extremal.{{refn | group=Note | name=ExtremalVsExtremum | Note the difference between the terms extremal and extremum. An extremal is a function that makes a functional an extremum.}} The extremum {{math|J [ f ]}} is called a local maximum if {{math|ΔJ ≤ 0}} everywhere in an arbitrarily small neighborhood of {{math|f ,}} and a local minimum if {{math|ΔJ ≥ 0}} there. For a function space of continuous functions, extrema of corresponding functionals are called weak extrema or strong extrema, depending on whether the first derivatives of the continuous functions are respectively all continuous or not.{{harvnb|Gelfand|Fomin|2000|pp=12–13}}Both strong and weak extrema of functionals are for a space of continuous functions but weak 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.{{harvnb | Gelfand|Fomin| 2000 | p=13 }} An example of a necessary condition that is used for finding weak extrema is the Euler–Lagrange equation.{{harvnb | Gelfand|Fomin| 2000 | pp=14–15 }} {{refn | group=Note | name=SectionVarSuffCond | For a sufficient condition, see section Variations and sufficient condition for a minimum.}}

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 where the functional derivative is equal to zero. This leads to solving the associated Euler–Lagrange equation.The following derivation of the Euler–Lagrange equation corresponds to the derivation on pp. 184–5 of:BOOK, Courant, R., Richard Courant, Hilbert, D., David Hilbert, Methods of Mathematical Physics, Vol. I, First English, Interscience Publishers, Inc., 1953, New York, 978-0471504474, Consider the functional
J[y] = int_{x_1}^{x_2} L(x,y(x),y'(x)), dx , .
where
{{math|x1, x2}} are constants, {{math|y (x)}} is twice continuously differentiable, {{math|y ′(x) {{=}} dy / dx  }}, {{math|L(x, y (x), y ′(x))}} is twice continuously differentiable with respect to its arguments {{math|x,  y,  y ′}}.
If the functional {{math|J[y ]}} attains a local minimum at {{math|f ,}} and {{math|η(x)}} is an arbitrary function that has at least one derivative and vanishes at the endpoints {{math|x1}} and {{math|x2 ,}} then for any number {{math|ε}} close to 0,
J[f] le J[f + varepsilon eta] , .
The term {{math|εη}} is called the variation of the function {{math|f}} and is denoted by {{math|δf .}}Substituting  {{math|f + εη}} for {{math|y}}  in the functional {{math|J[ y ] ,}} the result is a function of {{math|''ε}},
Phi(varepsilon) = J[f+varepsiloneta] , .
Since the functional {{math|J[ y ]}} has a minimum for {{math|y {{=}} f ,}} the function {{math|Φ(ε)}} has a minimum at {{math|ε {{=}} 0}} and thus,The product {{math|εΦ′(0)}} is called the first variation of the functional {{math|J}} and is denoted by {{math|δJ}}. Some references define the first variation differently by leaving out the {{math|ε}} factor.
Phi'(0) equiv left.frac{dPhi}{dvarepsilon}right|_{varepsilon = 0} = int_{x_1}^{x_2} left.frac{dL}{dvarepsilon}right|_{varepsilon = 0} dx = 0 , .
Taking the total derivative of {{math|L[x, y, y ′] ,}} where {{math|y {{=}} f + ε η}} and {{math|y ′ {{=}} f ′ + ε η′}} are functions of {{math|ε}} but {{math|x}} is not,
frac{dL}{dvarepsilon}=frac{partial L}{partial y}frac{dy}{dvarepsilon} + frac{partial L}{partial y'}frac{dy'}{dvarepsilon}and since  {{math|dy /dε {{=}} η}}  and  {{math|dy ′/dε {{=}} η' }},
frac{dL}{dvarepsilon}=frac{partial L}{partial y}eta + frac{partial L}{partial y'}eta'.Therefore,
begin{align}int_{x_1}^{x_2} left.frac{dL}{dvarepsilon}right|_{varepsilon = 0} dx
& = int_{x_1}^{x_2} left(frac{partial L}{partial f} eta + frac{partial L}{partial f'} eta'right), dx
& = int_{x_1}^{x_2} frac{partial L}{partial f} eta , dx + left.frac{partial L}{partial f'} eta right|_{x_1}^{x_2} - int_{x_1}^{x_2} eta frac{d}{dx}frac{partial L}{partial f'} , dx
& = int_{x_1}^{x_2} left(frac{partial L}{partial f} eta - eta frac{d}{dx}frac{partial L}{partial f'} right), dx
end{align}
where {{math|L[x, y, y ′] → L[x, f, f ′]}} when ε = 0 and we have used integration by parts on the second term. The last term vanishes because {{math|η {{=}} 0}} at {{math|x1}} and {{math|x2}} by definition. Also, as previously mentioned the left side of the equation is zero so that
int_{x_1}^{x_2} eta (x) left(frac{partial L}{partial f} - frac{d}{dx}frac{partial L}{partial f'} right) , dx = 0 , .
According to the fundamental lemma of calculus of variations, the part of the integrand in parentheses is zero, i.e.
frac{partial L}{partial f} -frac{d}{dx} frac{partial L}{partial f'}=0
which is called the Euler–Lagrange equation. The left hand side of this equation is called the functional derivative of {{math|J[f]}} and is denoted {{math|δJ/δf(x) .}}In general this gives a second-order ordinary differential equation which can be solved to obtain the extremal function {{math| f(x) .}} The Euler–Lagrange equation is a necessary, but not sufficient, condition for an extremum {{math|J[f]}}. A sufficient condition for a minimum is given in the section Variations and sufficient condition for a minimum.

Example

In order to illustrate this process, consider the problem of finding the extremal function {{math|y {{=}} f (x) ,}} which is the shortest curve that connects two points {{math|(x1, y1)}} and {{math|(x2, y2) .}} The arc length of the curve is given by
A[y] = int_{x_1}^{x_2} sqrt{1 + [ y'(x) ]^2} , dx , ,
with
y,'(x) = frac{dy}{dx} , , y_1=f(x_1) , , y_2=f(x_2) , .
The Euler–Lagrange equation will now be used to find the extremal function {{math|f (x)}} that minimizes the functional {{math|A[y ] .}}
frac{partial L}{partial f} -frac{d}{dx} frac{partial L}{partial f'}=0
with
L = sqrt{1 + [ f'(x) ]^2} , .
Since {{math|f}} does not appear explicitly in {{math|L ,}} the first term in the Euler–Lagrange equation vanishes for all {{math|f (x) }} and thus,
frac{d}{dx} frac{partial L}{partial f'} = 0 , .
Substituting for {{math|L}} and taking the derivative,
frac{d}{dx} frac{ f'(x) } {sqrt{1 + [ f'(x) ]^2}} = 0 , .
Thus
frac{f'(x)}{sqrt{1+[f'(x)]^2}} = c , ,
for some constant c. Then
frac{[f'(x)]^2}{1+[f'(x)]^2} = c^2 , ,
where
0le c^2


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