View source for History of variational principles in physics

{{Use dmy dates|date=January 2020}}

In [[physics]], a '''[[variational principle]]''' is an alternative method for determining the state or dynamics of a physical system, by identifying it as an extremum (minimum, maximum or saddle point) of a function or functional. Variational methods are exploited in many modern software to simulate matter and light.

Since the development of [[analytical mechanics]] in the 18th century, the fundamental equations of physics have usually been established in terms of [[Stationary-action principle|action principles]], where the variational principle is applied to the [[Action (physics)|action]] of a system in order to recover the fundamental equation of motion.

[[File:Classical mechanics roadmap.svg|thumb|The variational principles approach to classical mechanics was developed parallelly with Newtonian mechanics.]]

This article describes the historical development of such action principles and other variational methods applied in physics. See [[History of physics]] for an overview and [[Outline of the history of physics]] for related histories.

==Antiquity==
Variational principles are found among earlier ideas in [[surveying]] and [[optics]]. The [[rope stretchers]] of [[ancient Egypt]] stretched corded ropes between two points to measure the path which minimized the distance of separation, and [[Claudius Ptolemy]], in his [[Geographia (Ptolemy)|Geographia]] (Bk 1, Ch 2), emphasized that one must correct for "deviations from a straight course"; in [[ancient Greece]] [[Euclid]] states in his ''Catoptrica'' that, for the path of light reflecting from a mirror, the [[angle of incidence (optics)|angle of incidence]] equals the [[angle of reflection]]; and [[Hero of Alexandria]] later showed that this path was the shortest length and least time.<ref name=Kline1972>{{cite book|last=Kline|first=Morris|title=Mathematical Thought from Ancient to Modern Times|url=https://archive.org/details/mathematicalthou0000unse|url-access=registration|publisher=Oxford University Press|location=New York|year=1972|pages= [https://archive.org/details/mathematicalthou0000unse/page/167 167]–168|isbn=0-19-501496-0}}</ref>{{rp|580}}

==First variational principles==

=== Principle of virtual work ===
In the static analysis of objects under forces but fixed at [[mechanical equilibrium]], the [[principle of virtual work]] imagines tiny mathematical shifts away from equilibrium. Each shift does work—energy lost or gained—against the forces, but the sum of all these bits of virtual work must be zero. This principle was developed by [[Johann Bernoulli]] in a letter to [[Pierre Varignon]] in 1715, but never separately published.<ref name=Coopersmith2017/>{{rp|23}} [[Cornelius Lanczos]] uses a slightly different definition as the single postulate for all analytic mechanics, showing thereby the power of energy based variational principles over Newtonian mechanics.<ref name=Coopersmith2017/>{{rp|87}}

=== D'Alembert's principle ===
In 1743 [[Jean le Rond d'Alembert]] generalized the concept we now call [[virtual work]] to dynamical systems with rigid constraints, like rods or string under tension, a form that became known as the [[d'Alembert principle]].<ref>{{Cite book |last=Hankins |first=Thomas L. |title=Jean d'Alembert: science and the enlightenment |date=1990 |publisher=Gordon and Breach |isbn=978-2-88124-399-8 |series=Classics in the history and philosophy of science |location=New York Philadelphia London}}</ref>{{rp|190}} In the case of static (in equilibrium) rigid bodies without friction, the principle of virtual work says the net work of all applied forces (<math>F_i^{(a)}</math>) under variation of positions (<math>r_i</math>)  is zero:
:<math>\Sigma_i F_i^{(a)} \cdot \delta r_i = 0 </math>
A similar condition but valid for dynamics (systems in motion) introduces, for each force, the change in momentum <math>\dot{p}_i</math>:
:<math>\Sigma_i (F_i -\dot{p}_i)  \cdot \delta r_i = 0 </math>
which is d'Alembert's principle.<ref name=Goldstein3>{{Cite book |last1=Goldstein |first1=Herbert |title=Classical mechanics |last2=Poole |first2=Charles P. |last3=Safko |first3=John L. |date=2008 |publisher=Addison Wesley |isbn=978-0-201-65702-9 |edition=3. ed., [Nachdr.] |location=San Francisco Munich}}</ref>{{rp|17}}

=== Principle of least time ===
The earlier geometrical ideas in optics were generalized by [[Pierre de Fermat]], who, in the 17th century, refined the principle to "light travels between two given points along the path of shortest ''time''"; now known as the principle of least time or [[Fermat's principle]]. Fermat showed that principle predicts the observed law of [[refraction]]. His approach was metaphysical, arguing that Nature acts simply and economically.<ref name=Kline1972/>{{rp|580}}

=== The brachystochrone problem ===
[[File:Brachistochrone.gif|thumb|The brachistochrone problem. The path of the least time shown in red.]]
[[File:Path function 2.PNG|thumb|Techniques based on small variations in the path of motion grew out of analysis of the brachistochrone problem.]]
In 1696 [[Johann Bernoulli]] posed a puzzle to European mathematicians: derive a curve for motion of a frictionless bead falling between a higher and a lower point in the least possible time. He named the curve the "[[brachistochrone]]", (from ''brachystos'', "shortest", and ''chronos'', "time")<ref name=Goldstine>{{Cite book |last=Goldstine |first=Herman H. |date=1980 |title=A History of the Calculus of Variations from the 17th through the 19th Century |url=http://link.springer.com/10.1007/978-1-4613-8106-8 |series=Studies in the History of Mathematics and Physical Sciences |location=New York, NY |publisher=Springer New York |volume=5 |doi=10.1007/978-1-4613-8106-8 |isbn=978-1-4613-8108-2}}</ref>{{rp|31}} [[Isaac Newton]], [[Gottfried Wilhelm Leibniz]] and others contributed solutions, and in 1718 Johann Bernoulli published an analysis based on the solution created by his brother [[James Bernoulli]]. All of these works, especially the approach taken by the Bernoullis, involved reasoning about small deviations in the path taken by the falling bead. Thus this became the first application of the variational technique, albeit as a special-case rather than a general principle.<ref name=Goldstine/>{{rp|68}}

=== Principle of least action ===
{{Main | Maupertuis principle}}
In 1744<ref>{{Cite journal |last=de Maupertuis |first=P. L. M. |date=1744 |title=Accord de différentes loix de la nature qui avoient jusqu'ici paru incompatibles |url=https://fr.wikisource.org/wiki/Accord_de_diff%C3%A9rentes_loix_de_la_nature_qui_avoient_jusqu%E2%80%99ici_paru_incompatibles |journal=Mémoires de l'Académie Royale des Sciences de Paris |pages=417–426}}</ref> and 1746,<ref>{{Cite journal |last=de Mapertuis |first=M. |date=1746 |title=Les Loix du mouvement et du repos déduites d'un principe metaphysique |url=https://fr.wikisource.org/wiki/Les_Loix_du_mouvement_et_du_repos_d%C3%A9duites_d%E2%80%99un_principe_metaphysique |journal=Histoire de l'Académie Royale des Sciences et des Belles Lettres |pages=267–294}}</ref> [[Pierre Louis Maupertuis]] generalized Fermat's concept to mechanics,<ref name=Whittaker>{{Cite book |last=Whittaker |first=Edmund T. |title=A history of the theories of aether & electricity. 2: The modern theories, 1900 - 1926 |date=1989 |publisher=Dover Publ |isbn=978-0-486-26126-3 |edition=Repr |location=New York}}</ref>{{rp|97}} in the form of a [[Stationary-action principle|principle of least action]].  
Maupertuis argued metaphysically, he felt that "Nature is thrifty in all its actions", and applied the principle broadly:

{{Blockquote|The laws of movement and of rest deduced from this principle being precisely the same as those observed in nature, we can admire the application of it to all phenomena. The movement of animals, the vegetative growth of plants ... are only its consequences; and the spectacle of the universe becomes so much the grander, so much more beautiful, the worthier of its Author, when one knows that a small number of laws, most wisely established, suffice for all movements.|Pierre Louis Maupertuis<ref>Chris Davis. [http://www.idlex.freeserve.co.uk/idle/evolution/ref/leastact.html ''Idle theory''] {{webarchive|url=https://web.archive.org/web/20060615043538/http://www.idlex.freeserve.co.uk/idle/evolution/ref/leastact.html |date=2006-06-15 }} (1998)</ref>}}

This notion of Maupertuis, although somewhat deterministic today, does capture much of the essence of variational mechanics.

In application to physics, Maupertuis suggested that the quantity to be minimized was the product of the duration (time) of movement within a system by the [[action (physics)|action]]; his definitions of action varied with the problems he discussed.<ref name=Kline1972/>{{rp|581}}
One form he used was called "[[vis viva]]",

{{Equation box 1
|indent =:
|title='''Maupertuis' principle'''
|equation = <math>\delta \int 2T(t)  dt=0</math>
|border=2
|border colour = #0073CF
|background colour=#F5FFFA}}

which is the integral of twice what we now call the [[kinetic energy]] ''T'' of the system.

=== Euler's refinement ===
[[Leonhard Euler]] corresponded with Maupertuis from 1740 to 1744;<ref name=Kline1972/>{{rp|582}} in 1744 Euler proposed a refined formulation of the least action principle in 1744.<ref>Leonhard Euler, ''Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes.'' (1744) Bousquet, Lausanne & Geneva. 320 pages. Reprinted in ''Leonhardi Euleri Opera Omnia: Series I vol 24.'' (1952) C. Cartheodory (ed.) Orell Fuessli, Zurich. [http://math.dartmouth.edu/~euler/pages/E065.html scanned copy of complete text] at ''[http://math.dartmouth.edu/~euler/ The Euler Archive]'', Dartmouth.</ref> He writes<ref>Euler, ''Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes: [http://math.dartmouth.edu/~euler/docs/originals/E065h Additamentum II]'', Ibid.</ref>

:"Let the mass of the projectile be ''M'', and let its squared velocity resulting from its height be <math>v</math> while being moved over a distance ''ds''.   The body will have a momentum <math>M \sqrt{v}</math> that, when multiplied by the distance ''ds'', will give <math>M ds \sqrt{v}</math>, the momentum of the body integrated over the distance ''ds''.  Now I assert that the curve thus described by the body to be the curve (from among all other curves connecting the same endpoints) that minimizes <math>\int M ds \sqrt{v}</math> or, provided that  ''M'' is constant, <math>\int ds \sqrt{v}</math>."<ref group="Note">Original: "Sit massa corporis projecti ==''M'', ejusque, dum spatiolum == ''ds'' emetitur, celeritas debita altitudini == ''v''; erit quantitas motus corporis in hoc loco == <math>M\sqrt{v}</math> ; quae per ipsum spatiolum ''ds'' multiplicata, dabit <math>M\,ds\sqrt{v}</math> motum corporis collectivum per spatiolum ''ds''. Iam dico lineam a corpore descriptam ita fore comparatam, ut, inter omnes alias lineas iisdem terminis contentas, sit <math>\int M ds \sqrt{v}</math>, seu, ob M constans, <math>\int ds \sqrt{v}</math> minimum."</ref>

As Euler states, <math>\int M ds \sqrt{v}</math> is the integral of the momentum over distance traveled (note that here <math>v</math> contrary to usual notation denotes the ''squared'' velocity) which, in modern notation, equals the [[action (physics)#abbreviated action|abbreviated action]]:<ref name=Goldstein3/>{{rp|359}}

{{Equation box 1
|indent =:
|title='''Euler's principle'''
|equation = <math>\delta\int p\,dq=0</math>
|border=2
|border colour = #0073CF
|background colour=#F5FFFA}}

In rather general terms he wrote that "Since the fabric of the Universe is most perfect and is the work of a most wise Creator, nothing whatsoever takes place in the Universe in which some relation of maximum and minimum does not appear."

Euler continued to write on the topic; in his ''Reflexions sur quelques loix generales de la nature'' (1748), he called the quantity "effort". His expression corresponds to what we would now call [[potential energy]], so that his statement of least action in statics is equivalent to the principle that a system of bodies at rest will adopt a configuration that minimizes total potential energy.

=== Lagrangian mechanics ===
The first use of the term "method of variations" came in 1755 through the work of a young [[Joseph Louis Lagrange]]; Euler presented Lagrange's approach to the [[Prussian Academy of Sciences|Berlin Academy]] in 1756 as the "[[calculus of variations]]". Unlike Euler, Lagrange's approach was purely analytic rather than geometrical.  Lagrange introduced the idea of variation of entire curves or paths between the endpoints than of individual coordinates. For this he introduced a new form of a differential, written <math>\delta</math>, that acts on integrals rather than <math>d</math> acting on coordinates.<ref name=Goldstine/>{{rp|111}}  His notation continues to be used today.<ref name=Kline1972/>{{rp|583}}

=== Hamilton-Jacobi mechanics ===
{{Main|Hamilton's principle}}
{{See also| Hamilton–Jacobi equation}}
The variational principle was not used to derive the equations of motion until almost 75 years later, when [[William Rowan Hamilton]] in 1834 and 1835<ref>W.R. Hamilton, "On a General Method in Dynamics.", ''Philosophical Transactions of the Royal Society'' [http://www.emis.de/classics/Hamilton/GenMeth.pdf Part I (1834) p.247-308]; [http://www.emis.de/classics/Hamilton/SecEssay.pdf Part II (1835) p. 95-144]. (''From the collection [http://www.emis.de/classics/Hamilton/ Sir William Rowan Hamilton (1805-1865): Mathematical Papers] edited by David R. Wilkins, School of Mathematics, Trinity College, Dublin 2, Ireland. (2000); also reviewed as [http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/ On a General Method in Dynamics]'')</ref>  applied the variational principle to the Lagrangian function <math>L=T-V</math> (where ''T'' is the kinetic energy and ''V'' the potential energy of an object) to obtain what are now called the [[Euler–Lagrange equation]]s.  Hamilton believed his results were constrained by conservation of energy, which he called conservation of living force.<ref name=NakaneFraser/>{{rp|163}}

While few German scientists read English papers in this era, in 1836 the German mathematician [[Carl Gustav Jacobi]] read of Hamilton's work and immediately began new mathematical work, publishing ground breaking work on the variational principle in the following year.<ref>G.C.J. Jacobi, ''Vorlesungen über Dynamik, gehalten an der Universität Königsberg im Wintersemester 1842-1843''. A. Clebsch (ed.) (1866); Reimer; Berlin. 290 pages, available online [https://web.archive.org/web/20071122083808/http://math-doc.ujf-grenoble.fr/cgi-bin/oeitem?id=OE_JACOBI__8_1_0 Œuvres complètes volume '''8'''] at [https://web.archive.org/web/20081123234823/http://math-doc.ujf-grenoble.fr/OEUVRES/ Gallica-Math] from the [http://gallica.bnf.fr/ Gallica Bibliothèque nationale de France].</ref> Among Jacobi's results was the extension of Hamilton's method to time-dependent potentials (or "force functions" as they were known at that time).<ref name="NakaneFraser">{{Cite journal |last1=Nakane |first1=Michiyo |last2=Fraser |first2=Craig G. |date=2002 |title=The Early History of Hamilton-Jacobi Dynamics 1834–1837 |url=https://onlinelibrary.wiley.com/doi/10.1111/j.1600-0498.2002.tb00613.x |journal=Centaurus |language=en |volume=44 |issue=3–4 |pages=161–227 |doi=10.1111/j.1600-0498.2002.tb00613.x |pmid=17357243 |issn=0008-8994}}</ref>{{rp|201}}

===Extensions by Gauss and Hertz ===
Other extremal principles of [[classical mechanics]] were formulated, such as [[Carl Friedrich Gauss]]'s 1829 [[Gauss's principle of least constraint|principle of least constraint]] and its corollary, [[Heinrich Hertz]]'s 1896 [[Hertz principle of least curvature|principle of least curvature]].

== Action principle names ==
{{Anchor | action principle names}}
Action principles were developed by trial and error over three centuries; the names of the principles are not self-describing.<ref name=HancTaylorTuleja/> [[Richard Feynman]], through his PhD thesis<ref>{{Cite thesis |last=Feynman |first=Richard P. |title=Feynman's Thesis — A New Approach to Quantum Theory |chapter=THE PRINCIPLE OF LEAST ACTION IN QUANTUM MECHANICS |date=August 2005 |pages=1–69 |chapter-url=http://www.worldscientific.com/doi/abs/10.1142/9789812567635_0001 |isbn=978-981-256-366-8 |doi=10.1142/9789812567635_0001 |language=en}}</ref> and later through his reinvention of the undergraduate physics course, reinvigorated the field of variational principles in physics.<ref name=HancTaylorTuleja>{{Cite journal |last1=Hanc |first1=Jozef |last2=Taylor |first2=Edwin F. |last3=Tuleja |first3=Slavomir |date=2005-07-01 |title=Variational mechanics in one and two dimensions |url=https://pubs.aip.org/ajp/article/73/7/603/1056166/Variational-mechanics-in-one-and-two-dimensions |journal=American Journal of Physics |language=en |volume=73 |issue=7 |pages=603–610 |doi=10.1119/1.1848516 |bibcode=2005AmJPh..73..603H |issn=0002-9505}}</ref> In the process he upended the terminology. Feynman called Hamilton's principal function simply the "action" and Hamilton's principle he called "the principle of least action".<ref name=FeynmanII>{{Cite book |last=Feynman |first=Richard P. |url=https://www.feynmanlectures.caltech.edu/II_19.html |title=The Feynman lectures on physics. Volume 2: Mainly electromagnetism and matter |date=2011 |publisher=Basic Books |isbn=978-0-465-02494-0 |edition=The new millennium edition, paperback first published |location=New York}}</ref> The table below summarizes the key terminology found in modern physics literature.

{| class="wikitable"
|+ Action principle terminology
|-
! colspan="3"  | action !! colspan="3" |   principle
|-
! definition !! | historical name !! | modern name !! | definition !! | common name !! | Alternative name
|-
|<math>S = \int_{t_1}^{t_2} L\ dt</math> || Hamilton's principle function<ref name=Goldstein3/>{{rp|431}} || action<ref name=HancTaylorTuleja/><ref name=Goldstein3/>{{rp|359}}  ||<math>(\delta S)_T = 0</math> ||  Hamilton's principle<ref name=HancTaylorTuleja/><ref name=GrayKarlNovikov/><ref name=Goldstein3/> || Least action,<ref name=FeynmanII/><ref>{{Cite book |last1=Hand |first1=Louis N. |url=https://www.cambridge.org/core/product/identifier/9780511801662/type/book |title=Analytical Mechanics |last2=Finch |first2=Janet D. |date=1998-11-13 |publisher=Cambridge University Press |isbn=978-0-521-57572-0 |edition=1 |doi=10.1017/cbo9780511801662}}</ref>{{rp|46}} Stationary action <ref>{{Cite journal |last=Schwinger |first=Julian  |title=Relativistic Quantum Field Theory |url=https://www.nobelprize.org/prizes/physics/1965/schwinger/lecture/ |access-date=2023-11-19 |journal=Science |date=1966 |volume=153 |issue=3739 |pages=949–953 |doi=10.1126/science.153.3739.949 |pmid=17837239 |bibcode=1966Sci...153..949S |language=en-US}}</ref>
|-
|<math>W = \int_{x_1}^{x_2} mv\ ds</math> || action<ref name=Goldstein3/>{{rp|359}} || abbreviated<ref name=HancTaylorTuleja/><ref name=Goldstein3/>{{rp|359}} or Maupertuis<ref name=GrayKarlNovikov/> action ||<math>(\delta W)_E = 0</math>  ||  Maupertuis's  principle<ref name=GrayKarlNovikov/><ref name=HancTaylorTuleja/>   || Least action<ref name=Goldstein3/>{{rp|356}}
|-
|}
The notation <math>(\delta S)_T = 0</math> means variations on <math>S</math> with <math>T = t_2 - t_1</math> fixed;  <math>(\delta W)_E</math> means variation with constant energy.<ref name=GrayKarlNovikov>{{Cite journal |last1=Gray |first1=C G |last2=Karl |first2=G |last3=Novikov |first3=V A |date=2004-02-01 |title=Progress in classical and quantum variational principles |url=https://iopscience.iop.org/article/10.1088/0034-4885/67/2/R02 |journal=Reports on Progress in Physics |volume=67 |issue=2 |pages=159–208 |doi=10.1088/0034-4885/67/2/R02 |issn=0034-4885|arxiv=physics/0312071 |bibcode=2004RPPh...67..159G |s2cid=10822903 }}</ref> The abbreviated action is sometimes labeled <math>S_0</math>.<ref name=HancTaylorTuleja/> Some authors use "stationary action" or "least action" to mean any variational principle involving action.<ref name=Coopersmith2017>{{Cite book |last=Coopersmith |first=Jennifer |url=http://archive.org/details/lazyuniverseintr0000coop |title=The lazy universe: an introduction to the principle of least action |date=2017 |publisher=Oxford; New York, NY : Oxford University Press |isbn=978-0-19-874304-0}}</ref>{{rp|viii}}<ref name=YourgrauMandelstam1979/>{{rp|92}}

== Modern action principles ==

=== In relativity ===
{{main|Variational methods in general relativity}}
In 1915 [[David Hilbert]] applied variational principles to derive the [[Einstein field equations|gravitational field equations]] of [[general relativity]] in agreement with [[Albert Einstein]]'s derivation.<ref name=MehraEinsteinHilbert>{{Cite book |last=Mehra |first=Jagdish |chapter=Einstein, Hilbert, and the Theory of Gravitation |editor-last=Mehra |editor-first=Jagdish |title=The physicist's conception of nature |date=1987 |publisher=Reidel |isbn=978-90-277-2536-3 |edition=Reprint |location=Dordrecht}}</ref> (Einstein and Hilbert discussed Einstein's work on general relativity in person and letters throughout 1915.<ref name=Straumann>{{Cite book |last=Straumann |first=Norbert|title=General relativity|edition=2|publisher=Springer|location=Dordrecht}}</ref>) Hilbert's approach required accepting the variational principle as "axiomatic", a broadly accepted requirement today but questionable to the physicists of 1915. 
Hilbert's variations were based on what became known as the [[Einstein–Hilbert action]], given by
:<math>\mathcal{S}[g] =\frac{1}{2\kappa}\int  R \sqrt{-g} \, \mathrm{d}^4 x</math>,
where {{Math|''&kappa;''}} is [[Einstein gravitational constant]], <math>g=\det(g_{\alpha\beta})</math> is the determinant of a spacetime [[Lorentz metric]] and <math>R</math> is the [[scalar curvature]].

===In quantum mechanics===
{{See also | History of quantum mechanics}}
Variational principles played decisive roles at critical times in the development of quantum mechanics.

==== Sommerfeld's atom ====
{{Main | Bohr–Sommerfeld model}}
Following [[Max Planck]]'s proposal that quantum radiators explain the [[blackbody radiation]] spectrum and [[Albert Einstein]] hypothesis of quantum radiation to explain the [[photoelectric effect]], [[Niels Bohr]] proposed quantized energy levels for the orbits in his model of the atom, thereby explaining the [[Balmer series]] for absorption of radiation by atoms. However this hypothesis involved no mechanical model. [[Arnold Sommerfeld]] then showed that quantization of the action of orbits for Hydrogen predicted the Balmer series, complete with relativistic corrections leading to [[fine structure]] in spectral lines. However, this approach could not be extended to atoms with more electrons and, more fundamentally, the quantum hypothesis itself had no explanation from this classical mechanics solution.<ref name=YourgrauMandelstam1979/>{{rp|97}}

==== Schrödinger's equation ====
Combining Einstein's relativity and photoelectric effect results, [[De Broglie]] suggested that Sommerfeld's quantized action may relate to quantized wave effects; [[Erwin Schrödinger]] took up this idea, applying [[Hamilton's optico-mechanical analogy]] to connect the quantized action to [[Hamilton-Jacobi equations]] for the action. Hamilton's connection between light rays and light waves now became a connection between matter trajectories and de Broglie [[matter waves]].<ref name=YourgrauMandelstam1979/>{{rp|119}}  The resulting [[Schrödinger equation]] became the first successful quantum mechanics.

==== Dirac's quantum action ====
The work that built on Schrödinger's equation relied on analogies to [[Hamiltonian mechanics]].
In 1933 [[Paul Dirac]] published a paper seeking an alternative formulation based on [[Lagrangian mechanics]]. He was motivated by the power of the action principle and the relativistic invariance of the action itself.<ref>{{Cite book |last=Dirac |first=P. A. M. |chapter=The Lagrangian in Quantum Mechanics |date=Aug 2005 |title=Feynman's Thesis — A New Approach to Quantum Theory |chapter-url=http://www.worldscientific.com/doi/abs/10.1142/9789812567635_0003 |language=en |publisher=WORLD SCIENTIFIC |pages=111–119 |doi=10.1142/9789812567635_0003 |isbn=978-981-256-366-8}}</ref> Dirac was able to show that the wavefunctions probability amplitude at <math>x_1, t_2</math> was related to the amplitude at <math>x_2, t_2</math> through a complex exponential function of the action.<ref name=MerhaRechenberg>Mehra, Jagdish, and Rechenberg, Helmut. The Conceptual Completion and Extensions of Quantum Mechanics 1932-1941. Epilogue: Aspects of the Further Development of Quantum Theory 1942-1999: Subject Index: Volumes 1 to 6. Germany, Springer, 2001.</ref>{{rp|1025}}

==== Feynman's least action mechanics ====
{{Main | Path integral formulation }}
In 1942, nearly a decade after Dirac's work, [[Richard Feynman]] built a new quantum mechanics [[path integral formulation|formulation]] on the action principle. Feynman interpreted Dirac's formula as a physical recipe for the probability amplitude contributions from every possible path between <math>x_1, t_2</math> and <math>x_2, t_2</math>. These possibilities [[wave interference|interfere]]; constructive interference gives the paths with the most amplitude. In the classical limit with large values of action compared to <math>h</math>, the single classical path given by the action principle results.<ref name=MerhaRechenberg/>{{rp|1027}}

==== Schwinger's quantum action principle ====
{{Main | Schwinger's quantum action principle }}
In 1950, [[Julian Schwinger]] revisited Dirac's Lagrangian paper to develop the action principle in a different direction.<ref name=MerhaRechenberg/>{{rp|1082}} Unlike Feynman's focus on paths, Schwinger's approach was "differential" or local.

=== In particle physics ===
The [[Standard Model]] is defined in terms of a [[Lagrangian density]] that includes all known [[elementary particle]]s, the [[Higgs field]] and three of the [[fundamental interaction]]s ([[electromagnetism]], [[weak interaction]] and [[strong interaction]], not including gravitational interaction). Its formulation started in the 1970s and has successfully explained almost all experimental results related to microscopic physics.<ref>{{Cite journal |last1=Woithe |first1=Julia |last2=Wiener |first2=Gerfried J |last3=Van der Veken |first3=Frederik F |date=May 2017 |title=Let's have a coffee with the Standard Model of particle physics! |journal=Physics Education |volume=52 |issue=3 |page=034001 |doi=10.1088/1361-6552/aa5b25 |bibcode=2017PhyEd..52c4001W |issn=0031-9120|doi-access=free }}</ref>

== Teleology in action principles ==
The breadth of physical phenomena subject to study by action principles lead scientists from all centuries to view these concepts as especially fundamental; the connection of two points by paths lead some to suggest a "purpose" to the selection of one particular path.<ref name="Stöltzner1994">{{cite encyclopedia|last=Stöltzner|first=Michael|encyclopedia=Inside Versus Outside |editor1 = H. Atmanspacher | editor2 = G. J. Dalenoort |title = Action Principles and Teleology|series=Springer Series in Synergetics|year=1994|volume=63|location=Berlin|publisher=Springer|isbn=978-3-642-48649-4|pages=33–62 |doi=10.1007/978-3-642-48647-0_3}}</ref> This [[teleology|teleological]] viewpoint runs from the earliest physics through Fermat, Maupertuis, and on up to [[Max Planck]],  without, however, any scientific backing.<ref name=YourgrauMandelstam1979>{{Cite book |last1=Yourgrau |first1=Wolfgang |title=Variational principles in dynamics and quantum theory |last2=Mandelstam |first2=Stanley |date=1979 |publisher=Dover Publ |isbn=978-0-486-63773-0 |edition=Republ. of the 3rd ed., publ. in 1968 |series=Dover books on physics and chemistry |location=New York, NY}}</ref>{{rp|162}} The use of colorful language continues in the modern era with phrases like "Nature's command (to) ''Explore all paths!''"<ref>{{Cite journal |last1=Ogborn |first1=Jon |last2=Taylor |first2=Edwin F. |date=2005 |title=Quantum physics explains Newton's laws of motion |url=https://www.eftaylor.com/pub/QMtoNewtonsLaws.pdf |journal=Physics Education |volume=40 |issue=1 |page=26 |doi=10.1088/0031-9120/40/1/001|bibcode=2005PhyEd..40...26O |s2cid=250809103 }}</ref> or "It isn't that a particle takes the path of least action but that it smells all the paths in the neighborhood...".<ref name=FeynmanII/>{{rp|II:19}}

== Variational methods ==

=== Ritz's work on elasticity and waves ===
[[John William Strutt, 3rd Baron Rayleigh|Lord Rayleigh]] was the first to popularly adapt the variational principles for the search of [[eigenvalues and eigenvectors]] for the study of [[Elasticity (physics)|elasticity]] and classical [[wave]]s in his 1877 ''Theory of Sound''.<ref name=":0">{{Cite journal |last=Leissa |first=A.W. |date=2005 |title=The historical bases of the Rayleigh and Ritz methods |url=https://linkinghub.elsevier.com/retrieve/pii/S0022460X05000362 |journal=Journal of Sound and Vibration |language=en |volume=287 |issue=4–5 |pages=961–978 |doi=10.1016/j.jsv.2004.12.021|bibcode=2005JSV...287..961L }}</ref> The Rayleigh method allows approximation of the fundamental frequencies without full knowledge of the material composition and without the requirement of computational power.<ref name=":0" /> From 1903 to 1908, [[Walther Ritz]] introduced a series of improved methods for static and free vibration problems based on the optimization of an [[ansatz]] or trial function. Ritz demonstrated his use in the [[Euler–Bernoulli beam theory]] and the determination of [[Chladni Figures|Chladni figures]].<ref name=":0" />

For years, Ritz works were poorly cited in Western Europe and would only become popular after Ritz death in 1909.<ref name=":1">{{Cite journal |last1=Gander |first1=Martin J. |last2=Wanner |first2=Gerhard |date=2012 |title=From Euler, Ritz, and Galerkin to Modern Computing |journal=SIAM Review |volume=54 |issue=4 |pages=627–666 |doi=10.1137/100804036 |issn=0036-1445|url=https://archive-ouverte.unige.ch/unige:171273 }}</ref> In Russia, physicists like [[Ivan Bubnov]] (in 1913) and [[Boris Galerkin]] (in 1915) would rediscover and popularize some of Ritz's methods from 1908. In 1940, Georgii I. Petrov improved these approximations.<ref name=":1" /> These methods are now known under different names, including Bubnov–Galerkin, [[Petrov–Galerkin method|Petrov–Galerkin]] and Ritz–Galerkin methods.<ref name=":0" />

In 1911, Rayleigh complemented Ritz for his method for solving Chladni's problem, but complained for the lack of citation of his earlier work. However the similarity between Rayleigh's and Ritz's method has sometimes been challenged.<ref name=":0" /><ref>{{Cite journal |last=Ilanko |first=Sinniah |date=2009 |title=Comments on the historical bases of the Rayleigh and Ritz methods |url=https://linkinghub.elsevier.com/retrieve/pii/S0022460X08005476 |journal=Journal of Sound and Vibration |language=en |volume=319 |issue=1–2 |pages=731–733 |doi=10.1016/j.jsv.2008.06.001|bibcode=2009JSV...319..731I }}</ref><ref name=":1" /> Ritz's methods are sometimes referred as [[Rayleigh–Ritz method]] or simply [[Ritz method]], depending on the procedure.<ref name=":0" /><ref name=":1" /> Ritz's method led to the development of [[finite element method]] for the numerical solution of [[partial differential equation]]s in physics.<ref name=":1" />

=== For quantum systems ===
The variational method of Ritz would found his use [[quantum mechanics]] with the development of [[Hellmann–Feynman theorem]]. The theorem was first discussed by Schrödinger in 1926, the first proof was given by Paul Güttinger in 1932, and later rediscovered independently by [[Wolfgang Pauli]] and [[Hans Hellmann]] in 1933, and by Feynman in 1939.{{Citation needed|date=November 2023}}

In [[quantum chemistry]] and [[condensed matter physics]], [[Variational method (quantum mechanics)|variational methods]] were developed to study atoms, molecules, nuclei and solids under a quantum mechanical framework. Some of these include the use of Ritz methods for the determination of the spectra of the [[helium atom]], 1930 [[Hartree–Fock method]], 1964 [[density functional theory]] and [[variational Monte Carlo]] and 1992 [[density matrix renormalization group]] (DMRG).{{Citation needed|date=November 2023}}

=== Quantum algorithms ===
In 2014, variational principles were part of a hybrid strategy, called [[noisy intermediate-scale quantum era|noisy intermediate-scale quantum]] (NISQ) computing, to combine powerful but imperfect quantum computers coupled with classical computers.<ref>{{Cite journal |last1=Bharti |first1=Kishor |last2=Cervera-Lierta |first2=Alba |last3=Kyaw |first3=Thi Ha |last4=Haug |first4=Tobias |last5=Alperin-Lea |first5=Sumner |last6=Anand |first6=Abhinav |last7=Degroote |first7=Matthias |last8=Heimonen |first8=Hermanni |last9=Kottmann |first9=Jakob S. |last10=Menke |first10=Tim |last11=Mok |first11=Wai-Keong |last12=Sim |first12=Sukin |last13=Kwek |first13=Leong-Chuan |last14=Aspuru-Guzik |first14=Alán |date=2022-02-15 |title=Noisy intermediate-scale quantum algorithms |url=https://link.aps.org/doi/10.1103/RevModPhys.94.015004 |journal=Reviews of Modern Physics |language=en |volume=94 |issue=1 |page=015004 |doi=10.1103/RevModPhys.94.015004 |arxiv=2101.08448 |bibcode=2022RvMP...94a5004B |issn=0034-6861|hdl=10356/161272 |s2cid=231662441 |hdl-access=free }}</ref> The first proposals included a
[[variational quantum eigensolver]]<ref>{{Cite journal |last1=Peruzzo |first1=Alberto |last2=McClean |first2=Jarrod |last3=Shadbolt |first3=Peter |last4=Yung |first4=Man-Hong |last5=Zhou |first5=Xiao-Qi |last6=Love |first6=Peter J. |last7=Aspuru-Guzik |first7=Alán |last8=O’Brien |first8=Jeremy L. |date=2014-07-23 |title=A variational eigenvalue solver on a photonic quantum processor |journal=Nature Communications |language=en |volume=5 |issue=1 |page=4213 |doi=10.1038/ncomms5213 |issn=2041-1723 |pmc=4124861 |pmid=25055053|arxiv=1304.3061 |bibcode=2014NatCo...5.4213P }}</ref> to exploit quantum phenomena to simulate atoms and small molecules using variational methods and an approximate optimization algorithm.<ref>{{Cite arXiv |first1=Edward |last1=Farhi |first2=Jeffrey |last2=Goldstone |title=A Quantum Approximate Optimization Algorithm |date=2014-11-14 |last3=Gutmann |first3=Sam |class=quant-ph |eprint=1411.4028}}</ref><ref>{{Cite journal |last1=Blekos |first1=Kostas |last2=Brand |first2=Dean |last3=Ceschini |first3=Andrea |last4=Chou |first4=Chiao-Hui |last5=Li |first5=Rui-Hao |last6=Pandya |first6=Komal |last7=Summer |first7=Alessandro |date=2024-06-02 |title=A review on Quantum Approximate Optimization Algorithm and its variants |url=https://www.sciencedirect.com/science/article/pii/S0370157324001078 |journal=Physics Reports |volume=1068 |pages=1–66 |doi=10.1016/j.physrep.2024.03.002 |issn=0370-1573|arxiv=2306.09198 |bibcode=2024PhR..1068....1B }}</ref>

== Footnote ==
<references group="Note" />

==References==
<references/>

{{History of physics}}

{{DEFAULTSORT:History of Variational Principles in Physics}}
[[Category:Calculus of variations]]
[[Category:History of physics|Variational principles in physics]]