calculus of variations

The calculus of variations began with the work of Isaac Newton, such as with Newton's minimal resistance problem, which he formulated and solved in 1685, and later published in his Principia in 1687,{{Cite book |last=Goldstine |first=Herman H. |url=https://books.google.com/books?id=_iTnBwAAQBAJ&pg=PA7 |title=A History of the Calculus of Variations from the 17th Through the 19th Century |date=1980 |publisher=Springer New York |isbn=978-1-4613-8106-8 |series= |location= |pages=7–21}} 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.{{Citation |last=Ferguson |first=James |title=A Brief Survey of the History of the Calculus of Variations and its Applications |date=2004 |arxiv=math/0402357 |bibcode=2004math......2357F }}{{Cite book |last=Rowlands |first=Peter |url=https://books.google.com/books?id=ipA4DwAAQBAJ&pg=PA36 |title=Newton and the Great World System |date=2017 |publisher=World Scientific Publishing |isbn=978-1-78634-372-7 |pages=36–39 |language=en |doi=10.1142/q0108}}{{Cite journal |last=Torres |first=Delfim F. M. |date=2021-07-29 |title=On a Non-Newtonian Calculus of Variations |journal=Axioms |language=en |volume=10 |issue=3 |pages=171 |doi=10.3390/axioms10030171 |doi-access=free |issn=2075-1680|arxiv=2107.14152 }} This problem was followed by the brachistochrone curve problem raised by Johann Bernoulli (1696),{{cite book| last1=Gelfand|first1=I. M.|author-link1=Israel Gelfand|last2=Fomin|first2=S. V.|author-link2=Sergei Fomin|title=Calculus of variations | year=2000|publisher=Dover Publications|location=Mineola, New York|isbn=978-0486414485|page=3| url=https://books.google.com/books?id=YkFLGQeGRw4C|edition=Unabridged repr.|editor1-last=Silverman| editor1-first=Richard A.}} which was similar to one raised by Galileo Galilei 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 and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733. Joseph-Louis Lagrange was influenced by Euler's work to contribute greatly 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.{{cite book |last=Thiele |first=Rüdiger |editor-last1=Bradley |editor-first1=Robert E. |editor-last2=Sandifer |editor-first2=C. Edward |title=Leonhard Euler: Life, Work and Legacy |publisher=Elsevier |year=2007 |page=249 |chapter=Euler and the Calculus of Variations |chapter-url=https://books.google.com/books?id=75vJL_Y-PvsC&pg=PA249 |isbn=9780080471297}}{{cite book |last=Goldstine |first=Herman H. |year=2012 |title=A History of the Calculus of Variations from the 17th through the 19th Century |url=https://books.google.com/books?id=_iTnBwAAQBAJ&q=%22Indeed+after%22&pg=110 |publisher=Springer Science & Business Media |page=110 |isbn=9781461381068 |author-link=Herman Goldstine }}{{efn|"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."}}

Adrien-Marie 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.{{cite book |last=van Brunt |first=Bruce |title=The Calculus of Variations |publisher=Springer |year=2004 |isbn=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 Pierre Frédéric Sarrus (1842) which was condensed and improved by Augustin-Louis Cauchy (1844). Other valuable treatises and memoirs have been written by Strauch{{which|date=October 2024}} (1849), John Hewitt 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 Karl 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, Oskar Bolza, Gilbert Ames Bliss, 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.{{cite arXiv |last=Ferguson |first=James |eprint=math/0402357 |title= Brief Survey of the History of the Calculus of Variations and its Applications |year=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.{{cite journal |last=Bellman |first=Richard E. |title= Dynamic Programming and a new formalism in the calculus of variations |year=1954 |journal= Proc. Natl. Acad. Sci. | issue=4 | pages=231–235|pmc=527981 |pmid=16589462 |volume=40 |doi=10.1073/pnas.40.4.231|bibcode=1954PNAS...40..231B |doi-access=free }}{{cite web |title=Richard E. Bellman Control Heritage Award |year=2004 |url=http://a2c2.org/awards/richard-e-bellman-control-heritage-award |work=American Automatic Control Council |access-date=2013-07-28 |archive-date=2018-10-01 |archive-url=https://web.archive.org/web/20181001032837/http://a2c2.org/awards/richard-e-bellman-control-heritage-award |url-status=dead }}{{efn|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 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 defined over a given domain. A functional $J[y]$ is said to have an extremum at the function $f$ if $\backslash Delta\; J\; =\; J[y]\; -\; J[f]$ has the same sign for all $y$ in an arbitrarily small neighborhood of $f.${{efn|The neighborhood of $f$ is the part of the given function space where over the whole domain of the functions, with $h$ a positive number that specifies the size of the neighborhood.{{cite book |last1=Courant |first1=R |author-link1=Richard Courant |last2=Hilbert |first2=D |author-link2=David Hilbert |title = Methods of Mathematical Physics |volume=I |edition=First English |publisher=Interscience Publishers, Inc. |year=1953 |location=New York |page=169 |isbn=978-0471504474}}}} The function $f$ is called an extremal function or extremal.{{efn|name=ExtremalVsExtremum| Note the difference between the terms extremal and extremum. An extremal is a function that makes a functional an extremum.}} The extremum $J[f]$ is called a local maximum if $\backslash Delta\; J\; \backslash leq\; 0$ everywhere in an arbitrarily small neighborhood of $f,$ and a local minimum if $\backslash Delta\; J\; \backslash geq\; 0$ 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.{{harvnb|Gelfand|Fomin|2000|pp=12–13}}

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.{{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 }}{{efn|name=SectionVarSuffCond| For a sufficient condition, see section Variations and sufficient condition for a minimum.}}

{{main|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.{{efn|The following derivation of the Euler–Lagrange equation corresponds to the derivation on pp. 184–185 of Courant & Hilbert (1953).{{cite book |author=Courant, R. |author-link=Richard Courant |author2=Hilbert, D. |author2-link= David Hilbert |title=Methods of Mathematical Physics |volume=I |edition=First English |publisher=Interscience Publishers, Inc. |year=1953 |location=New York |isbn=978-0471504474}}}}

Consider the functional

$$J[y]\; =\; \backslash int\_\{x\_1\}^\{x\_2\}\; L\backslash left(x,y(x),y\text{'}(x)\backslash right)\backslash ,\; dx\; \backslash ,\; .$$

where

• $x\_1,\; x\_2$ are constants,
• $y(x)$ is twice continuously differentiable,
• $y\text{'}(x)\; =\; \backslash frac\{dy\}\{dx\},$
• $L\backslash left(x,\; y(x),\; y\text{'}(x)\backslash right)$ is twice continuously differentiable with respect to its arguments $x,\; y,$ and $y\text{'}.$

If the functional $J[y]$ attains a local minimum at $f,$ and $\backslash eta(x)$ is an arbitrary function that has at least one derivative and vanishes at the endpoints $x\_1$ and $x\_2,$ then for any number $\backslash varepsilon$ close to 0,

$$J[f]\; \backslash le\; J[f\; +\; \backslash varepsilon\; \backslash eta]\; \backslash ,\; .$$

The term $\backslash varepsilon\; \backslash eta$ is called the variation of the function $f$ and is denoted by $\backslash delta\; f.${{efn|Note that $\backslash eta(x)$ and $f(x)$ are evaluated at the {{em|same}} values of $x,$ which is not valid more generally in variational calculus with non-holonomic constraints.}}

Substituting $f\; +\; \backslash varepsilon\; \backslash eta$ for $y$ in the functional $J[y],$ the result is a function of $\backslash varepsilon,$

$$\backslash Phi(\backslash varepsilon)\; =\; J[f+\backslash varepsilon\backslash eta]\; \backslash ,\; .$$

Since the functional $J[y]$ has a minimum for $y\; =\; f$ the function $\backslash Phi(\backslash varepsilon)$ has a minimum at $\backslash varepsilon\; =\; 0$ and thus,{{efn|The product $\backslash varepsilon\; \backslash Phi\text{'}(0)$ is called the first variation of the functional $J$ and is denoted by $\backslash delta\; J.$ Some references define the first variation differently by leaving out the $\backslash varepsilon$ factor.}}

$$\backslash Phi\text{'}(0)\; \backslash equiv\; \backslash left.\backslash frac\{d\backslash Phi\}\{d\backslash varepsilon\}\backslash right|\_\{\backslash varepsilon\; =\; 0\}\; =\; \backslash int\_\{x\_1\}^\{x\_2\}\; \backslash left.\backslash frac\{dL\}\{d\backslash varepsilon\}\backslash right|\_\{\backslash varepsilon\; =\; 0\}\; dx\; =\; 0\; \backslash ,\; .$$

Taking the total derivative of $L\backslash left[x,\; y,\; y\text{'}\backslash right],$ where $y\; =\; f\; +\; \backslash varepsilon\; \backslash eta$ and $y\text{'}\; =\; f\text{'}\; +\; \backslash varepsilon\; \backslash eta\text{'}$ are considered as functions of $\backslash varepsilon$ rather than $x,$ yields

$$\backslash frac\{dL\}\{d\backslash varepsilon\}=\backslash frac\{\backslash partial\; L\}\{\backslash partial\; y\}\backslash frac\{dy\}\{d\backslash varepsilon\}\; +\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; y\text{'}\}\backslash frac\{dy\text{'}\}\{d\backslash varepsilon\}$$

and because $\backslash frac\{dy\}\{d\; \backslash varepsilon\}\; =\; \backslash eta$ and $\backslash frac\{d\; y\text{'}\}\{d\; \backslash varepsilon\}\; =\; \backslash eta\text{'},$

$$\backslash frac\{dL\}\{d\backslash varepsilon\}=\backslash frac\{\backslash partial\; L\}\{\backslash partial\; y\}\backslash eta\; +\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; y\text{'}\}\backslash eta\text{'}.$$

Therefore,

$$\backslash begin\{align\}$$

\int_{x_1}^{x_2} \left.\frac{dL}{d\varepsilon}\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 $L\backslash left[x,\; y,\; y\text{'}\backslash right]\; \backslash to\; L\backslash left[x,\; f,\; f\text{'}\backslash right]$ when $\backslash varepsilon\; =\; 0$ and we have used integration by parts on the second term. The second term on the second line vanishes because $\backslash eta\; =\; 0$ at $x\_1$ and $x\_2$ by definition. Also, as previously mentioned the left side of the equation is zero so that

$$\backslash int\_\{x\_1\}^\{x\_2\}\; \backslash eta\; (x)\; \backslash left(\backslash frac\{\backslash partial\; L\}\{\backslash partial\; f\}\; -\; \backslash frac\{d\}\{dx\}\backslash frac\{\backslash partial\; L\}\{\backslash partial\; f\text{'}\}\; \backslash right)\; \backslash ,\; dx\; =\; 0\; \backslash ,\; .$$

According to the fundamental lemma of calculus of variations, the part of the integrand in parentheses is zero, i.e.

$$\backslash frac\{\backslash partial\; L\}\{\backslash partial\; f\}\; -\backslash frac\{d\}\{dx\}\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; f\text{'}\}=0$$

which is called the Euler–Lagrange equation. The left hand side of this equation is called the functional derivative of $J[f]$ and is denoted $\backslash delta\; J$ or $\backslash delta\; f(x).$

In general this gives a second-order ordinary differential equation which can be solved to obtain the extremal function $f(x).$ The Euler–Lagrange equation is a necessary, but not sufficient, condition for an extremum $J[f].$ A sufficient condition for a minimum is given in the section Variations and sufficient condition for a minimum.

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 $\backslash left(x\_1,\; y\_1\backslash right)$ and $\backslash left(x\_2,\; y\_2\backslash right).$ The arc length of the curve is given by

$$A[y]\; =\; \backslash int\_\{x\_1\}^\{x\_2\}\; \backslash sqrt\{1\; +\; [\; y\text{'}(x)\; ]^2\}\; \backslash ,\; dx\; \backslash ,\; ,$$

with

$$y\text{'}(x)\; =\; \backslash frac\{dy\}\{dx\}\; \backslash ,\; ,\; \backslash \; \backslash \; y\_1=f(x\_1)\; \backslash ,\; ,\; \backslash \; \backslash \; y\_2=f(x\_2)\; \backslash ,\; .$$

Note that assuming {{mvar|y}} is a function of {{mvar|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 $f(x)$ that minimizes the functional $A[y].$

$$\backslash frac\{\backslash partial\; L\}\{\backslash partial\; f\}\; -\backslash frac\{d\}\{dx\}\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; f\text{'}\}=0$$

with

$$L\; =\; \backslash sqrt\{1\; +\; [\; f\text{'}(x)\; ]^2\}\; \backslash ,\; .$$

Since $f$ does not appear explicitly in $L,$ the first term in the Euler–Lagrange equation vanishes for all $f(x)$ and thus,

$$\backslash frac\{d\}\{dx\}\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; f\text{'}\}\; =\; 0\; \backslash ,\; .$$

Substituting for $L$ and taking the derivative,

$$\backslash frac\{d\}\{dx\}\; \backslash \; \backslash frac\{f\text{'}(x)\}\; \{\backslash sqrt\{1\; +\; [f\text{'}(x)]^2\}\}\; \backslash \; =\; 0\; \backslash ,\; .$$

Thus

$$\backslash frac\{f\text{'}(x)\}\{\backslash sqrt\{1+[f\text{'}(x)]^2\}\}\; =\; c\; \backslash ,\; ,$$

for some constant $c.$ Then

$$\backslash frac\{[f\text{'}(x)]^2\}\{1+[f\text{'}(x)]^2\}\; =\; c^2\; \backslash ,\; ,$$

where

Solving, we get

$$[f\text{'}(x)]^2=\backslash frac\{c^2\}\{1-c^2\}$$

which implies that

$$f\text{'}(x)=m$$

is a constant and therefore that the shortest curve that connects two points $\backslash left(x\_1,\; y\_1\backslash right)$ and $\backslash left(x\_2,\; y\_2\backslash right)$ is

$$f(x)\; =\; m\; x\; +\; b\; \backslash qquad\; \backslash text\{with\}\; \backslash \; \backslash \; m\; =\; \backslash frac\{y\_2\; -\; y\_1\}\{x\_2\; -\; x\_1\}\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; b\; =\; \backslash frac\{x\_2\; y\_1\; -\; x\_1\; y\_2\}\{x\_2\; -\; x\_1\}$$

and we have thus found the extremal function $f(x)$ that minimizes the functional $A[y]$ so that $A[f]$ is a minimum. The equation for a straight line is $y\; =\; mx+b.$ In other words, the shortest distance between two points is a straight line.{{efn|name=ArchimedesStraight| As a historical note, this is an axiom of Archimedes. See e.g. Kelland (1843).{{cite book |last=Kelland |first=Philip |author-link=Philip Kelland| title=Lectures on the principles of demonstrative mathematics |year=1843 |page=58 |url=https://books.google.com/books?id=yQCFAAAAIAAJ&pg=PA58 |via=Google Books}}}}

In physics problems it may be the case that $\backslash frac\{\backslash partial\; L\}\{\backslash partial\; x\}\; =\; 0,$ meaning the integrand is a function of $f(x)$ and $f\text{'}(x)$ but $x$ does not appear separately. In that case, the Euler–Lagrange equation can be simplified to the Beltrami identity{{cite web |author=Weisstein, Eric W. | url=http://mathworld.wolfram.com/Euler-LagrangeDifferentialEquation.html |title=Euler–Lagrange Differential Equation | website=mathworld.wolfram.com |publisher=Wolfram |at=Eq. (5)}}

$$L\; -\; f\text{'}\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; f\text{'}\}\; =\; C\; \backslash ,\; ,$$

where $C$ is a constant. The left hand side is the Legendre transformation of $L$ with respect to $f\text{'}(x).$

The intuition behind this result is that, if the variable $x$ is actually time, then the statement $\backslash frac\{\backslash partial\; L\}\{\backslash partial\; x\}\; =\; 0$ implies that the Lagrangian is time-independent. By Noether's theorem, there is an associated conserved quantity. In this case, this quantity is the Hamiltonian, the Legendre transform of the Lagrangian, which (often) coincides with the energy of the system. This is (minus) the constant in Beltrami's identity.

If $S$ depends on higher-derivatives of $y(x),$ that is, if $$S\; =\; \backslash int\_\{a\}^\{b\}\; f(x,\; y(x),\; y\text{'}(x),\; \backslash dots,\; y^\{(n)\}(x))\; dx,$$ then $y$ must satisfy the Euler–Poisson equation,{{Cite book |last=Kot |first=Mark |title=A First Course in the Calculus of Variations | publisher=American Mathematical Society | year=2014 |isbn=978-1-4704-1495-5 | chapter=Chapter 4: Basic Generalizations}} $$\backslash frac\{\backslash partial\; f\}\{\backslash partial\; y\}\; -\; \backslash frac\{d\}\{dx\}\; \backslash left(\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; y\text{'}\}\; \backslash right)\; +\; \backslash dots\; +\; (-1)^\{n\}\; \backslash frac\{d^n\}\{dx^n\}\; \backslash left[\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; y^\{(n)\}\}\; \backslash right]=\; 0.$$

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

$$\backslash frac\{\backslash partial^2\; L\}\{\backslash partial\; f\text{'}^2\}\; \backslash ne\; 0,$$

then $f$ has two continuous derivatives, and it satisfies the Euler–Lagrange equation.

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:{{Cite journal|last=Manià|first=Bernard|date=1934|title=Sopra un esempio di Lavrentieff| journal=Bollenttino dell'Unione Matematica Italiana|volume=13|pages=147–153}}

$$L[x]\; =\; \backslash int\_0^1\; (x^3-t)^2\; x\text{'}^6,$$

$$\{A\}\; =\; \backslash \{x\; \backslash in\; W^\{1,1\}(0,1)\; :\; x(0)=0,\backslash \; x(1)=1\backslash \}.$$

Clearly, $x(t)\; =\; t^\{\backslash frac\{1\}\{3\}\}$minimizes the functional, but we find any function $x\; \backslash in\; W^\{1,\; \backslash infty\}$ gives a value bounded away from the infimum.

Examples (in one-dimension) are traditionally manifested across $W^\{1,1\}$ and $W^\{1,\backslash infty\},$ but Ball and Mizel{{Cite journal|last=Ball & Mizel|date=1985|title=One-dimensional Variational problems whose Minimizers do not satisfy the Euler-Lagrange equation.|journal=Archive for Rational Mechanics and Analysis|volume=90|issue=4|pages=325–388| doi=10.1007/BF00276295|bibcode=1985ArRMA..90..325B|s2cid=55005550}} procured the first functional that displayed Lavrentiev's Phenomenon across $W^\{1,p\}$ and $W^\{1,q\}$ 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.{{Cite journal|last=Ferriero|first=Alessandro|date=2007|title=The Weak Repulsion property | journal=Journal de Mathématiques Pures et Appliquées|volume=88|issue=4|pages=378–388| doi=10.1016/j.matpur.2007.06.002 | doi-access=}}

For example, if $\backslash varphi(x,\; y)$ 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:

$$U[\backslash varphi]\; =\; \backslash iint\_D\; \backslash sqrt\{1\; +\backslash nabla\; \backslash varphi\; \backslash cdot\; \backslash nabla\; \backslash varphi\}\; \backslash ,dx\backslash ,dy.$$

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:

$$\backslash varphi\_\{xx\}(1\; +\; \backslash varphi\_y^2)\; +\; \backslash varphi\_\{yy\}(1\; +\; \backslash varphi\_x^2)\; -\; 2\backslash varphi\_x\; \backslash varphi\_y\; \backslash varphi\_\{xy\}\; =\; 0.$$

See Courant (1950) for details.

It is often sufficient to consider only small displacements of the membrane, whose energy difference from no displacement is approximated by

$$V[\backslash varphi]\; =\; \backslash frac\{1\}\{2\}\backslash iint\_D\; \backslash nabla\; \backslash varphi\; \backslash cdot\; \backslash nabla\; \backslash varphi\; \backslash ,\; dx\backslash ,\; dy.$$

The functional $V$ is to be minimized among all trial functions $\backslash varphi$ 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 $V[u\; +\; \backslash varepsilon\; v]$ must vanish:

$$\backslash left.\backslash frac\{d\}\{d\backslash varepsilon\}\; V[u\; +\; \backslash varepsilon\; v]\backslash right|\_\{\backslash varepsilon=0\}\; =\; \backslash iint\_D\; \backslash nabla\; u\; \backslash cdot\; \backslash nabla\; v\; \backslash ,\; dx\backslash ,dy\; =\; 0.$$

Provided that u has two derivatives, we may apply the divergence theorem to obtain

$$\backslash iint\_D\; \backslash nabla\; \backslash cdot\; (v\; \backslash nabla\; u)\; \backslash ,dx\backslash ,dy\; =$$

\iint_D \nabla u \cdot \nabla v + v \nabla \cdot \nabla u \,dx\,dy = \int_C v \frac{\partial u}{\partial n} \, ds,

where $C$ is the boundary of $D,$ $s$ is arclength along $C$ and $\backslash partial\; u\; /\; \backslash partial\; n$ is the normal derivative of $u$ on $C.$ Since $v$ vanishes on $C$ and the first variation vanishes, the result is

$$\backslash iint\_D\; v\backslash nabla\; \backslash cdot\; \backslash nabla\; u\; \backslash ,dx\backslash ,dy\; =0$$

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

$$\backslash nabla\; \backslash cdot\; \backslash nabla\; u=\; 0$$in $D.$

The difficulty with this reasoning is the assumption that the minimizing function $u$ must have two derivatives. Riemann argued that the existence of a smooth minimizing function was assured by the connection with the physical problem: membranes do indeed assume configurations with minimal potential energy. Riemann named this idea the Dirichlet principle in honor of his teacher Peter Gustav Lejeune Dirichlet. However Weierstrass gave an example of a variational problem with no solution: minimize

$$W[\backslash varphi]\; =\; \backslash int\_\{-1\}^\{1\}\; (x\backslash varphi\text{'})^2\; \backslash ,\; dx$$

among all functions $\backslash varphi$ that satisfy $\backslash varphi(-1)=-1$ and $\backslash varphi(1)=1.$

$W$ 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 $W=0.${{efn|The resulting controversy over the validity of Dirichlet's principle is explained by Turnbull.{{cite web |url=http://turnbull.mcs.st-and.ac.uk/~history/Biographies/Riemann.html |title=Riemann biography |publisher=U. St. Andrew |place=UK |author=Turnbull}}}} 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).

A more general expression for the potential energy of a membrane is

$$V[\backslash varphi]\; =\; \backslash iint\_D\; \backslash left[\; \backslash frac\{1\}\{2\}\; \backslash nabla\; \backslash varphi\; \backslash cdot\; \backslash nabla\; \backslash varphi\; +\; f(x,y)\; \backslash varphi\; \backslash right]\; \backslash ,\; dx\backslash ,dy\; \backslash ,\; +\; \backslash int\_C\; \backslash left[\; \backslash frac\{1\}\{2\}\; \backslash sigma(s)\; \backslash varphi^2\; +\; g(s)\; \backslash varphi\; \backslash right]\; \backslash ,\; ds.$$

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 $\backslash sigma(s)$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 $V[u\; +\; \backslash varepsilon\; v]$ is given by

$$\backslash iint\_D\; \backslash left[\; \backslash nabla\; u\; \backslash cdot\; \backslash nabla\; v\; +\; f\; v\; \backslash right]\; \backslash ,\; dx\backslash ,\; dy\; +\; \backslash int\_C\; \backslash left[\; \backslash sigma\; u\; v\; +\; g\; v\; \backslash right]\; \backslash ,\; ds\; =\; 0.$$

If we apply the divergence theorem, the result is

$$\backslash iint\_D\; \backslash left[\; -v\; \backslash nabla\; \backslash cdot\; \backslash nabla\; u\; +\; v\; f\; \backslash right]\; \backslash ,\; dx\; \backslash ,\; dy\; +\; \backslash int\_C\; v\; \backslash left[\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; n\}\; +\; \backslash sigma\; u\; +\; g\; \backslash right]\; \backslash ,\; ds\; =0.$$

If we first set $v\; =\; 0$ on $C,$ the boundary integral vanishes, and we conclude as before that

$$-\; \backslash nabla\; \backslash cdot\; \backslash nabla\; u\; +\; f\; =0$$

in $D.$ Then if we allow $v$ to assume arbitrary boundary values, this implies that $u$ must satisfy the boundary condition

$$\backslash frac\{\backslash partial\; u\}\{\backslash partial\; n\}\; +\; \backslash sigma\; u\; +\; g\; =0,$$

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 $\backslash sigma$ vanishes identically on $C.$ In such a case, we could allow a trial function $\backslash varphi\; \backslash equiv\; c,$ where $c$ is a constant. For such a trial function,

$$V[c]\; =\; c\backslash left[\; \backslash iint\_D\; f\; \backslash ,\; dx\backslash ,dy\; +\; \backslash int\_C\; g\; \backslash ,\; ds\; \backslash right].$$

By appropriate choice of $c,$ $V$ can assume any value unless the quantity inside the brackets vanishes. Therefore, the variational problem is meaningless unless

$$\backslash iint\_D\; f\; \backslash ,\; dx\backslash ,dy\; +\; \backslash int\_C\; g\; \backslash ,\; ds\; =0.$$

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).

Both one-dimensional and multi-dimensional eigenvalue problems can be formulated as variational problems.

{{See also|Sturm–Liouville theory}}

The Sturm–Liouville eigenvalue problem involves a general quadratic form

$$Q[y]\; =\; \backslash int\_\{x\_1\}^\{x\_2\}\; \backslash left[\; p(x)\; y\text{'}(x)^2\; +\; q(x)\; y(x)^2\; \backslash right]\; \backslash ,\; dx,$$

where $y$ is restricted to functions that satisfy the boundary conditions

$$y(x\_1)=0,\; \backslash quad\; y(x\_2)=0.$$

Let $R$ be a normalization integral

$$R[y]\; =\backslash int\_\{x\_1\}^\{x\_2\}\; r(x)y(x)^2\; \backslash ,\; dx.$$

The functions $p(x)$ and $r(x)$ are required to be everywhere positive and bounded away from zero. The primary variational problem is to minimize the ratio $Q/R$ among all $y$ satisfying the endpoint conditions, which is equivalent to minimizing $Q[y]$ under the constraint that $R[y]$ is constant. It is shown below that the Euler–Lagrange equation for the minimizing $u$ is

$$-(p\; u\text{'})\text{'}\; +q\; u\; -\backslash lambda\; r\; u\; =\; 0,$$

where $\backslash lambda$ is the quotient

$$\backslash lambda\; =\; \backslash frac\{Q[u]\}\{R[u]\}.$$

It can be shown (see Gelfand and Fomin 1963) that the minimizing $u$ has two derivatives and satisfies the Euler–Lagrange equation. The associated $\backslash lambda$ will be denoted by $\backslash lambda\_1$; 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

$$\backslash int\_\{x\_1\}^\{x\_2\}\; r(x)\; u\_1(x)\; y(x)\; \backslash ,\; dx\; =\; 0.$$

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

$$Q[y]\; =\; \backslash int\_\{x\_1\}^\{x\_2\}\; \backslash left[\; p(x)\; y\text{'}(x)^2\; +\; q(x)y(x)^2\; \backslash right]\; \backslash ,\; dx\; +\; a\_1\; y(x\_1)^2\; +\; a\_2\; y(x\_2)^2,$$

where $a\_1$ and $a\_2$ are arbitrary. If we set $y\; =\; u\; +\; \backslash varepsilon\; v$, the first variation for the ratio $Q/R$ is

$$V\_1\; =\; \backslash frac\{2\}\{R[u]\}\; \backslash left(\; \backslash int\_\{x\_1\}^\{x\_2\}\; \backslash left[\; p(x)\; u\text{'}(x)v\text{'}(x)\; +\; q(x)u(x)v(x)\; -\backslash lambda\; r(x)\; u(x)\; v(x)\; \backslash right]\; \backslash ,\; dx\; +\; a\_1\; u(x\_1)v(x\_1)\; +\; a\_2\; u(x\_2)v(x\_2)\; \backslash right),$$

where λ is given by the ratio $Q[u]/R[u]$ as previously.

After integration by parts,

$$\backslash frac\{R[u]\}\{2\}\; V\_1\; =\; \backslash int\_\{x\_1\}^\{x\_2\}\; v(x)\; \backslash left[\; -(p\; u\text{'})\text{'}\; +\; q\; u\; -\backslash lambda\; r\; u\; \backslash right]\; \backslash ,\; dx\; +\; v(x\_1)[\; -p(x\_1)u\text{'}(x\_1)\; +\; a\_1\; u(x\_1)]\; +\; v(x\_2)\; [p(x\_2)\; u\text{'}(x\_2)\; +\; a\_2\; u(x\_2)].$$

If we first require that $v$ vanish at the endpoints, the first variation will vanish for all such $v$ only if

If $u$ satisfies this condition, then the first variation will vanish for arbitrary $v$ only if

$$-p(x\_1)u\text{'}(x\_1)\; +\; a\_1\; u(x\_1)=0,\; \backslash quad\; \backslash hbox\{and\}\; \backslash quad\; p(x\_2)\; u\text{'}(x\_2)\; +\; a\_2\; u(x\_2)=0.$$

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 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

$$Q[\backslash varphi]\; =\; \backslash iiint\_D\; p(X)\; \backslash nabla\; \backslash varphi\; \backslash cdot\; \backslash nabla\; \backslash varphi\; +\; q(X)\; \backslash varphi^2\; \backslash ,\; dx\; \backslash ,\; dy\; \backslash ,\; dz\; +\; \backslash iint\_B\; \backslash sigma(S)\; \backslash varphi^2\; \backslash ,\; dS,$$

and

$$R[\backslash varphi]\; =\; \backslash iiint\_D\; r(X)\; \backslash varphi(X)^2\; \backslash ,\; dx\; \backslash ,\; dy\; \backslash ,\; dz.$$

Let $u$ be the function that minimizes the quotient $Q[\backslash varphi]\; /\; R[\backslash varphi],$

with no condition prescribed on the boundary $B.$ The Euler–Lagrange equation satisfied by $u$ is

$$-\backslash nabla\; \backslash cdot\; (p(X)\; \backslash nabla\; u)\; +\; q(x)\; u\; -\; \backslash lambda\; r(x)\; u=0,$$

where

$$\backslash lambda\; =\; \backslash frac\{Q[u]\}\{R[u]\}.$$

The minimizing $u$ must also satisfy the natural boundary condition

$$p(S)\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; n\}\; +\; \backslash sigma(S)\; u\; =\; 0,$$

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).

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

$$A[f]\; =\; \backslash int\_\{x\_0\}^\{x\_1\}\; n(x,f(x))\; \backslash sqrt\{1\; +\; f\text{'}(x)^2\}\; dx,$$

where the refractive index $n(x,y)$ depends upon the material.

If we try $f(x)\; =\; f\_0\; (x)\; +\; \backslash varepsilon\; f\_1\; (x)$ then the first variation of $A$ (the derivative of $A$ with respect to ε) is

$$\backslash delta\; A[f\_0,f\_1]\; =\; \backslash int\_\{x\_0\}^\{x\_1\}\; \backslash left[\; \backslash frac\{\; n(x,f\_0)\; f\_0\text{'}(x)\; f\_1\text{'}(x)\}\{\backslash sqrt\{1\; +\; f\_0\text{'}(x)^2\}\}\; +\; n\_y\; (x,f\_0)\; f\_1\; \backslash sqrt\{1\; +\; f\_0\text{'}(x)^2\}\; \backslash right]\; dx.$$

After integration by parts of the first term within brackets, we obtain the Euler–Lagrange equation

$$-\backslash frac\{d\}\{dx\}\; \backslash left[\backslash frac\{\; n(x,f\_0)\; f\_0\text{'}\}\{\backslash sqrt\{1\; +\; f\_0\text{'}^2\}\}\; \backslash right]\; +\; n\_y\; (x,f\_0)\; \backslash sqrt\{1\; +\; f\_0\text{'}(x)^2\}\; =\; 0.$$

The light rays may be determined by integrating this equation. This formalism is used in the context of Lagrangian optics and Hamiltonian optics.

There is a discontinuity of the refractive index when light enters or leaves a lens. Let

$$n(x,y)\; =\; \backslash begin\{cases\}$$

n_{(-)} & \text{if} \quad x<0, \\

n_{(+)} & \text{if} \quad x>0,

\end{cases}

where $n\_\{(-)\}$ and $n\_\{(+)\}$ are constants. Then the Euler–Lagrange equation holds as before in the region where 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 $f\text{'}$ may be discontinuous. After integration by parts in the separate regions and using the Euler–Lagrange equations, the first variation takes the form

$$\backslash delta\; A[f\_0,f\_1]\; =\; f\_1(0)\backslash left[\; n\_\{(-)\}\backslash frac\{f\_0\text{'}(0^-)\}\{\backslash sqrt\{1\; +\; f\_0\text{'}(0^-)^2\}\}\; -\; n\_\{(+)\}\backslash frac\{f\_0\text{'}(0^+)\}\{\backslash sqrt\{1\; +\; f\_0\text{'}(0^+)^2\}\}\; \backslash right].$$

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.

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 $\backslash dot\; X(t)$ be its tangent vector. The optical length of the curve is given by

$$A[C]\; =\; \backslash int\_\{t\_0\}^\{t\_1\}\; n(X)\; \backslash sqrt\{\; \backslash dot\; X\; \backslash cdot\; \backslash dot\; X\}\; \backslash ,\; dt.$$

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

$$\backslash frac\{d\}\{dt\}\; P\; =\; \backslash sqrt\{\; \backslash dot\; X\; \backslash cdot\; \backslash dot\; X\}\; \backslash ,\; \backslash nabla\; n,$$

where

$$P\; =\; \backslash frac\{n(X)\; \backslash dot\; X\}\{\backslash sqrt\{\backslash dot\; X\; \backslash cdot\; \backslash dot\; X\}\; \}.$$

It follows from the definition that $P$ satisfies

$$P\; \backslash cdot\; P\; =\; n(X)^2.$$

Therefore, the integral may also be written as

$$A[C]\; =\; \backslash int\_\{t\_0\}^\{t\_1\}\; P\; \backslash cdot\; \backslash dot\; X\; \backslash ,\; dt.$$

This form suggests that if we can find a function $\backslash psi$ whose gradient is given by $P,$ then the integral $A$ is given by the difference of $\backslash psi$ 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 $\backslash psi.$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.

The wave equation for an inhomogeneous medium is

$$u\_\{tt\}\; =\; c^2\; \backslash nabla\; \backslash cdot\; \backslash nabla\; u,$$

where $c$ is the velocity, which generally depends upon $X.$ Wave fronts for light are characteristic surfaces for this partial differential equation: they satisfy

$$\backslash varphi\_t^2\; =\; c(X)^2\; \backslash ,\; \backslash nabla\; \backslash varphi\; \backslash cdot\; \backslash nabla\; \backslash varphi.$$

We may look for solutions in the form

$$\backslash varphi(t,X)\; =\; t\; -\; \backslash psi(X).$$

In that case, $\backslash psi$ satisfies

$$\backslash nabla\; \backslash psi\; \backslash cdot\; \backslash nabla\; \backslash psi\; =\; n^2,$$

where $n=1/c.$ According to the theory of first-order partial differential equations, if $P\; =\; \backslash nabla\; \backslash psi,$ then $P$ satisfies

$$\backslash frac\{dP\}\{ds\}\; =\; n\; \backslash ,\; \backslash nabla\; n,$$

along a system of curves (the light rays) that are given by

$$\backslash frac\{dX\}\{ds\}\; =\; P.$$

These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification

$$\backslash frac\{ds\}\{dt\}\; =\; \backslash frac\{\backslash sqrt\{\; \backslash dot\; X\; \backslash cdot\; \backslash dot\; X\}\; \}\{n\}.$$

We conclude that the function $\backslash psi$ 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.

{{main|Action (physics)}}

In classical mechanics, the action, $S,$ is defined as the time integral of the Lagrangian, $L.$ The Lagrangian is the difference of energies,

$$L\; =\; T\; -\; U,$$

where $T$ is the kinetic energy of a mechanical system and $U$ its potential energy. 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

$$S\; =\; \backslash int\_\{t\_0\}^\{t\_1\}\; L(x,\; \backslash dot\; x,\; t)\; \backslash ,\; dt$$

is stationary with respect to variations in the path $x(t).$

The Euler–Lagrange equations for this system are known as Lagrange's equations:

$$\backslash frac\{d\}\{dt\}\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\; x\}\; =\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; x\},$$

and they are equivalent to Newton's equations of motion (for such systems).

The conjugate momenta $P$ are defined by

$$p\; =\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\; x\}.$$

For example, if

$$T\; =\; \backslash frac\{1\}\{2\}\; m\; \backslash dot\; x^2,$$

then $$p\; =\; m\; \backslash dot\; x.$$

Hamiltonian mechanics results if the conjugate momenta are introduced in place of $\backslash dot\; x$ by a Legendre transformation of the Lagrangian $L$ into the Hamiltonian $H$ defined by

$$H(x,\; p,\; t)\; =\; p\; \backslash ,\backslash dot\; x\; -\; L(x,\backslash dot\; x,\; t).$$

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:

$$\backslash frac\{\backslash partial\; \backslash psi\}\{\backslash partial\; t\}\; +\; H\backslash left(x,\backslash frac\{\backslash partial\; \backslash psi\}\{\backslash partial\; x\},t\backslash right)\; =\; 0.$$

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.

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{{efn|name=AltFirst| The first variation is also called the variation, differential, or first differential.}} is defined as the linear part of the change in the functional, and the second variation{{efn|name=AltSecond| The second variation is also called the second differential.}} is defined as the quadratic part.{{harvnb|Gelfand|Fomin|2000|pp=11–12, 99}}

For example, if $J[y]$ is a functional with the function $y\; =\; y(x)$ 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{{efn|name=SimplifyNotation|Note that $\backslash Delta\; J[h]$ and the variations below, depend on both $y$ and $h.$ The argument $y$ has been left out to simplify the notation. For example, $\backslash Delta\; J[h]$ could have been written $\backslash Delta\; J[y;\; h].${{harvnb | Gelfand|Fomin|2000 | p=12, footnote 6}}}}

$$\backslash Delta\; J[h]\; =\; J[y+h]\; -\; J[y].$$

The functional $J[y]$ is said to be differentiable if

$$\backslash Delta\; J[h]\; =\; \backslash varphi\; [h]\; +\; \backslash varepsilon\; \backslash |h\backslash |,$$

where $\backslash varphi[h]$ is a linear functional,{{efn|name=Linear|A functional $\backslash varphi[h]$ is said to be linear if $\backslash varphi[\backslash alpha\; h]\; =\; \backslash alpha\; \backslash varphi[h]$   and   $\backslash varphi\backslash left[h\; +\; h\_2\backslash right]\; =\; \backslash varphi[h]\; +\; \backslash varphi\backslash left[h\_2\backslash right],$ where $h,\; h\_2$ are functions and $\backslash alpha$ is a real number.{{harvnb | Gelfand|Fomin| 2000 | p=8 }}}} $\backslash |h\backslash |$ is the norm of $h,${{efn|name=Norm| For a function $h\; =\; h(x)$ that is defined for $a\; \backslash leq\; x\; \backslash leq\; b,$ where $a$ and $b$ are real numbers, the norm of $h$ is its maximum absolute value, i.e. $\backslash |h\backslash |\; =\; \backslash displaystyle\backslash max\_\{a\; \backslash leq\; x\; \backslash leq\; b\}\; |h(x)|.${{harvnb | Gelfand|Fomin| 2000 | p=6 }}}} and $\backslash varepsilon\; \backslash to\; 0$ as $\backslash |h\backslash |\; \backslash to\; 0.$ The linear functional $\backslash varphi[h]$ is the first variation of $J[y]$ and is denoted by,{{harvnb | Gelfand|Fomin| 2000 | pp=11–12}}

$$\backslash delta\; J[h]\; =\; \backslash varphi[h].$$

The functional $J[y]$ is said to be twice differentiable if

$$\backslash Delta\; J[h]\; =\; \backslash varphi\_1\; [h]\; +\; \backslash varphi\_2\; [h]\; +\; \backslash varepsilon\; \backslash |h\backslash |^2,$$

where $\backslash varphi\_1[h]$ is a linear functional (the first variation), $\backslash varphi\_2[h]$ is a quadratic functional,{{efn|name=Quadratic| 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.{{harvnb | Gelfand|Fomin| 2000 | pp=97–98 }}}} and $\backslash varepsilon\; \backslash to\; 0$ as $\backslash |h\backslash |\; \backslash to\; 0.$ The quadratic functional $\backslash varphi\_2[h]$ is the second variation of $J[y]$ and is denoted by,{{harvnb | Gelfand|Fomin| 2000 | p=99 }}

$$\backslash delta^2\; J[h]\; =\; \backslash varphi\_2[h].$$

The second variation $\backslash delta^2\; J[h]$ is said to be strongly positive if

$$\backslash delta^2J[h]\; \backslash ge\; k\; \backslash |h\backslash |^2,$$

for all $h$ and for some constant $k\; >\; 0$.{{harvnb | Gelfand|Fomin| 2000 | p=100 }}

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.

{{quote box|align=left |fontsize=100% |border=2px |quote=Sufficient condition for a minimum:

{{block indent | em = 1.5 | text = The functional $J[y]$ has a minimum at $y\; =\; \backslash hat\{y\}$ if its first variation $\backslash delta\; J[h]\; =\; 0$ at $y\; =\; \backslash hat\{y\}$ and its second variation $\backslash delta^2\; J[h]$ is strongly positive at $y\; =\; \backslash hat\{y\}.${{harvnb | Gelfand|Fomin| 2000 | p=100 |loc=Theorem 2}}{{Efn|name=sufficient| For other sufficient conditions, see in {{harvnb|Gelfand|Fomin|2000}},

• Chapter{{nbsp}}5: "The Second Variation. Sufficient Conditions for a Weak Extremum" – Sufficient conditions for a weak minimum are given by the theorem on p.{{nbsp}}116.
• Chapter{{nbsp}}6: "Fields. Sufficient Conditions for a Strong Extremum" – Sufficient conditions for a strong minimum are given by the theorem on p.{{nbsp}}148.}}{{efn|name=FuncMin| 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.}} }}}}

{{clear}}

{{Div col|colwidth=25em}}

• First 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
• {{cat|Variational analysts}}
• Measures of central tendency as solutions to variational problems
• Stampacchia Medal
• Fermat Prize
• Convenient vector space
• Variational vector field

{{div col end}}

{{notelist|1}}

{{reflist|25em}}

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• Bolza, O.: [http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ACM2513 Lectures on the Calculus of Variations]. Chelsea Publishing Company, 1904, available on Digital Mathematics library. 2nd edition republished in 1961, paperback in 2005, {{isbn|978-1-4181-8201-4}}.
• Cassel, Kevin W.: [https://books.google.com/books?id=pNHFJTKixisC Variational Methods with Applications in Science and Engineering], Cambridge University Press, 2013.
• Clegg, J.C.: [https://books.google.com/books/about/Calculus_of_variations.html?id=qTnvAAAAMAAJ Calculus of Variations], Interscience Publishers Inc., 1968.
• Courant, R.: [https://books.google.com/books?id=GCDSBwAAQBAJ Dirichlet's principle, conformal mapping and minimal surfaces]. Interscience, 1950.
• Dacorogna, Bernard: "[http://www.worldscientific.com/doi/suppl/10.1142/p967/suppl_file/p967_chap01.pdf Introduction]" [http://www.worldscientific.com/worldscibooks/10.1142/p967 Introduction to the Calculus of Variations], 3rd edition. 2014, World Scientific Publishing, {{isbn|978-1-78326-551-0}}.
• Elsgolc, L.E.: [https://books.google.com/books?id=MAU_AwAAQBAJ Calculus of Variations], Pergamon Press Ltd., 1962.
• Forsyth, A.R.: [https://archive.org/stream/CalculusOfVariations/Forsyth-CalculusOfVariations#page/n3/mode/2up Calculus of Variations], Dover, 1960.
• Fox, Charles: [https://books.google.com/books/about/An_Introduction_to_the_Calculus_of_Varia.html?id=PV2IovCWTcYC An Introduction to the Calculus of Variations], Dover Publ., 1987.
• Giaquinta, Mariano; Hildebrandt, Stefan: Calculus of Variations I and II, Springer-Verlag, {{ISBN|978-3-662-03278-7}} and {{ISBN|978-3-662-06201-2}}
• Jost, J. and X. Li-Jost: [https://books.google.com/books?id=QN8Iw7fUA-8C Calculus of Variations]. Cambridge University Press, 1998.
• Lebedev, L.P. and Cloud, M.J.: [https://books.google.com/books?id=_T3ez-32YVsC The Calculus of Variations and Functional Analysis with Optimal Control and Applications in Mechanics], World Scientific, 2003, pages 1–98.
• Logan, J. David: [https://books.google.com/books?id=nUk_AQAAIAAJ Applied Mathematics], 3rd edition. Wiley-Interscience, 2006
• {{cite book|chapter-url=http://www.mpri.lsu.edu/textbook/Chapter8-b.htm|chapter=Chapter 8: Calculus of Variations| url=http://www.mpri.lsu.edu/bookindex.html|title=Optimization for Engineering Systems| first=Ralph W. |last=Pike |publisher=Louisiana State University|archive-url=https://web.archive.org/web/20070705141725/http://www.mpri.lsu.edu/bookindex.html|url-status=dead|archive-date=2007-07-05}}
• Roubicek, T.: "[https://web.archive.org/web/20150510023928/http://www.wiley-vch.de/books/sample/3527411887_c17.pdf Calculus of variations]". Chap.17 in: [https://web.archive.org/web/20150510021514/http://www.wiley-vch.de/publish/en/books/forthcomingTitles/MA00/3-527-41188-7/?sID=nrgsqk516u2v9ffab8u7io1dq4 Mathematical Tools for Physicists]. (Ed. M. Grinfeld) J. Wiley, Weinheim, 2014, {{isbn|978-3-527-41188-7}}, pp. 551–588.
• Sagan, Hans: [https://books.google.com/books/about/Introduction_to_the_Calculus_of_Variatio.html?id=abhS8PgpBskC Introduction to the Calculus of Variations], Dover, 1992.
• Weinstock, Robert: [https://books.google.com/books?id=6wSVuWH1PrsC Calculus of Variations with Applications to Physics and Engineering], Dover, 1974 (reprint of 1952 ed.).

• [https://www.encyclopediaofmath.org/index.php/Variational_calculus Variational calculus]. Encyclopedia of Mathematics.
• [http://planetmath.org/calculusofvariations calculus of variations]. PlanetMath.
• [http://mathworld.wolfram.com/CalculusofVariations.html Calculus of Variations]. MathWorld.
• [https://web.archive.org/web/20170609215523/http://www.exampleproblems.com/wiki/index.php/Calculus_of_Variations Calculus of variations]. Example problems.
• [https://www.youtube.com/playlist?list=PL521C2DFD15FF568C Mathematics - Calculus of Variations and Integral Equations]. Lectures on YouTube.
• Selected papers on Geodesic Fields. [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/geodesic_fields_-_pt._1.pdf Part I], [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/geodesic_fields_-_pt._2.pdf Part II].

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