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{{Short description|Scientific principles enabling the use of the calculus of variations}}{{Inline references needed|date=November 2023}}{{Calculus|expanded=specialized}}

A '''variational principle''' is a mathematical procedure that renders a physical problem solvable by the [[calculus of variations]], which concerns finding functions that optimize the values of quantities that depend on those functions. For example, the problem of determining the shape of a hanging chain suspended at both ends—a [[catenary]]—can be solved using [[variational calculus]], and in this case, the variational principle is the following: The solution is a function that minimizes the [[gravitational energy|gravitational potential energy]] of the chain.

==History==

===Physics===
{{main|History of variational principles in physics}}
The history of the variational principle in [[classical mechanics]] started with [[Maupertuis's principle]] in the 18th century.
===Math===
[[Felix Klein]]'s 1872 [[Erlangen program]] attempted to identify invariants under a [[Group (mathematics)|group]] of transformations.

==Examples==
===In mathematics===
* [[Ekeland's variational principle]] in mathematical optimization
* The [[finite element method]]
* The variation principle relating [[topological entropy]] and [[Kolmogorov-Sinai entropy]].

=== In physics ===
[[File:Classical_mechanics_roadmap.svg|thumb|Variational principles for classical mechanics was developed in parallel with Newtonian mechanics.]]
* The [[Rayleigh–Ritz method]] for solving [[Boundary value problem|boundary-value problem]]s in elasticity and wave propagation
* [[Fermat's principle]] in [[geometrical optics]]
* [[Hamilton's principle]] in [[classical mechanics]]
* [[Maupertuis' principle]] in [[classical mechanics]]
* The [[principle of least action]] in [[mechanics]], [[electromagnetic theory]], and [[quantum mechanics]]
* The [[variational method (quantum mechanics)|variational method]] in quantum mechanics
* [[Hellmann–Feynman theorem]]
* [[Gauss's principle of least constraint]] and [[Gauss's principle of least constraint#Hertz's principle of least curvature|Hertz's principle of least curvature]]
* [[Einstein–Hilbert action|Hilbert's action principle]] in general relativity, leading to the [[Einstein field equations]].
* [[Palatini variation]]
* [[Hartree–Fock method]]
* [[Density functional theory]]
* [[Gibbons–Hawking–York boundary term]]
* [[Variational quantum eigensolver]]

==References==
{{Reflist}}

==External links==
*[https://feynmanlectures.caltech.edu/II_19.html The Feynman Lectures on Physics Vol. II Ch. 19: The Principle of Least Action]
* {{cite journal|last=Ekeland|first=Ivar|authorlink=Ivar Ekeland|title=Nonconvex minimization problems|journal=Bulletin of the American Mathematical Society|series=New Series|volume=1|year=1979|number=3|pages=443–474|doi=10.1090/S0273-0979-1979-14595-6|mr=526967|doi-access=free}}
* S T Epstein  1974 "The Variation Method in Quantum Chemistry". (New York: Academic)
* C Lanczos, ''The Variational Principles of Mechanics'' (Dover Publications)
* R K Nesbet 2003 "Variational Principles and Methods In Theoretical Physics and Chemistry". (New York: Cambridge U.P.)
* S K Adhikari  1998 "Variational Principles for the Numerical Solution of Scattering Problems". (New York: Wiley)
* C G Gray, G Karl G and V A Novikov 1996, ''Ann. Phys.'' 251 1.
* C.G. Gray, G. Karl, and V. A. Novikov, "[https://arxiv.org/abs/physics/0312071 Progress in Classical and Quantum Variational Principles]". 11 December 2003. physics/0312071 Classical Physics.
*{{cite book |author=Griffiths, David J. |title=Introduction to Quantum Mechanics (2nd ed.) |publisher=Prentice Hall |year=2004 |isbn=0-13-805326-X |url-access=registration |url=https://archive.org/details/introductiontoel00grif_0 }}
* John Venables, "[http://venables.asu.edu/quant/varprin.html The Variational Principle and some applications] {{Webarchive|url=https://web.archive.org/web/20150926212853/http://venables.asu.edu/quant/varprin.html |date=2015-09-26 }}". Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona (Graduate Course: Quantum Physics)
* Andrew James Williamson, "[http://www.tcm.phy.cam.ac.uk/~ajw29/thesis/node15.html The Variational Principle] -- Quantum monte carlo calculations of electronic excitations". Robinson College, Cambridge,  Theory of Condensed Matter Group, Cavendish Laboratory. September 1996. (dissertation of Doctor of Philosophy)
* Kiyohisa Tokunaga, "[https://web.archive.org/web/20041102095610/http://www.d3.dion.ne.jp/~kiyohisa/tieca/26.htm Variational Principle for Electromagnetic Field]". Total Integral for Electromagnetic Canonical Action, Part Two, Relativistic Canonical Theory of Electromagnetics, Chapter VI
*[[Vadim Komkov|Komkov, Vadim]] (1986) Variational principles of continuum mechanics with engineering applications. Vol. 1. Critical points theory. Mathematics and its Applications, 24. D. Reidel Publishing Co., Dordrecht.
* Cassel, Kevin W.: Variational Methods with Applications in Science and Engineering, Cambridge University Press, 2013.

[[Category:Calculus of variations| ]]
[[Category:Theoretical physics]]
[[Category:Articles containing proofs]]
[[Category:Principles]]
[[Category:Variational principles| ]]