A variational principle in physics 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. This article describes the historical development of such principles.
Before modern times
Variational principles are found among earlier ideas in
surveying
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...
and
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
. 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
Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importance ...
, in his
Geographia
The ''Geography'' ( grc-gre, Γεωγραφικὴ Ὑφήγησις, ''Geōgraphikḕ Hyphḗgēsis'', "Geographical Guidance"), also known by its Latin names as the ' and the ', is a gazetteer, an atlas, and a treatise on cartography, com ...
(Bk 1, Ch 2), emphasized that one must correct for "deviations from a straight course"; in
ancient Greece
Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cult ...
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
states in his ''Catoptrica'' that, for the path of light reflecting from a mirror, the
angle of incidence equals the
angle of reflection
Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves. The ' ...
; and
Hero of Alexandria
Hero of Alexandria (; grc-gre, Ἥρων ὁ Ἀλεξανδρεύς, ''Heron ho Alexandreus'', also known as Heron of Alexandria ; 60 AD) was a Greece, Greek mathematician and engineer who was active in his native city of Alexandria, Roman Egy ...
later showed that this path was the shortest length and least time.
This was generalized to
refraction
In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomeno ...
by
Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
, 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
Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the pat ...
or
Fermat's principle
Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the pat ...
.
Principle of extremal action
Credit for the formulation of the principle of least action is commonly given to
Pierre Louis Maupertuis
Pierre Louis Moreau de Maupertuis (; ; 1698 – 27 July 1759) was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Prussian Academy of Science, at the ...
, who wrote about it in 1744 and 1746, although the true priority is less clear, as discussed below.
Maupertuis felt that "Nature is thrifty in all its actions", and applied the principle broadly: "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."
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 "
vis viva
''Vis viva'' (from the Latin for "living force") is a historical term used for the first recorded description of what we now call kinetic energy in an early formulation of the principle of conservation of energy.
Overview
Proposed by Gottfried L ...
", twice what we now call the kinetic energy of the system.
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
gave a formulation of the action principle in 1744, in very recognizable terms, in the ''Additamentum 2'' to his "Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes". He begins the second paragraph:
:"Sit massa corporis projecti ''M'', ejusque, dum spatiolum ''ds'' emetitur, celeritas debita altitudini ''v''; erit quantitas motus corporis in hoc loco
; quae per ipsum spatiolum ''ds'' multiplicata, dabit
motum corporis collectivum per spatiolum ''ds''. Iam dico lineam a corpore descriptam ita fore comparatam, ut, inter omnes alias lineas iisdem terminis contentas, sit
, seu, ob M constans,
minimum."
A translation of this passage reads:
:"Let the mass of the projectile be ''M'', and let its squared velocity resulting from its height be
while being moved over a distance ''ds''. The body will have a momentum
that, when multiplied by the distance ''ds'', will give
, 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
or, provided that ''M'' is constant,
."
As Euler states,
is the integral of the momentum over distance traveled (note that here
contrary to usual notation denotes the ''squared'' velocity) which, in modern notation, equals the
reduced action . Thus, Euler made an equivalent and (apparently) independent statement of the variational principle in the same year as Maupertuis, albeit slightly later. 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."
However, Euler did not claim any priority, as the following episode shows.
Maupertuis' priority was disputed in 1751 by the mathematician
Samuel König
Samuel ''Šəmūʾēl'', Tiberian Hebrew, Tiberian: ''Šămūʾēl''; ar, شموئيل or صموئيل '; el, Σαμουήλ ''Samouḗl''; la, Samūēl is a figure who, in the narratives of the Hebrew Bible, plays a key role in the transit ...
, who claimed that it had been invented by
Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
in 1707. Although similar to many of Leibniz's arguments, the principle itself has not been documented in Leibniz's works. König himself showed a ''copy'' of a 1707 letter from Leibniz to
Jacob Hermann with the principle, but the ''original'' letter has been lost. In contentious proceedings, König was accused of forgery, and even the King of Prussia entered the debate, defending Maupertuis, while
Voltaire
François-Marie Arouet (; 21 November 169430 May 1778) was a French Age of Enlightenment, Enlightenment writer, historian, and philosopher. Known by his ''Pen name, nom de plume'' M. de Voltaire (; also ; ), he was famous for his wit, and his ...
defended König. Euler, rather than claiming priority, was a staunch defender of Maupertuis, and Euler himself prosecuted König for forgery before the Berlin Academy on 13 April 1752. The claims of forgery were re-examined 150 years later, and archival work by
C.I. Gerhardt in 1898 and
W. Kabitz in 1913 uncovered other copies of the letter, and three others cited by König, in the
Bernoulli Bernoulli can refer to:
People
*Bernoulli family of 17th and 18th century Swiss mathematicians:
** Daniel Bernoulli (1700–1782), developer of Bernoulli's principle
**Jacob Bernoulli (1654–1705), also known as Jacques, after whom Bernoulli numbe ...
archives.
Further developments
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
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potentia ...
, 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.
The full importance of the principle to mechanics was stated by
Joseph Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia[William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...](_blank)
in 1834 and 1835 applied the variational principle to the function
to obtain what are now called the
Lagrangian equations of motion
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lou ...
.
Alternative formulations
In 1842,
Carl Gustav Jacobi
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, Dynamics (mechanics), dynamics, differential equations, determinants, and number theory. H ...
tackled the problem of whether the variational principle found minima or other extrema (e.g. a
saddle point
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function ...
); most of his work focused on geodesics on two-dimensional surfaces. The first clear general statements were given by
Marston Morse
Harold Calvin Marston Morse (March 24, 1892 – June 22, 1977) was an American mathematician best known for his work on the ''calculus of variations in the large'', a subject where he introduced the technique of differential topology now known a ...
in the 1920s and 1930s, leading to what is now known as
Morse theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiabl ...
. For example, Morse showed that the number of conjugate points in a trajectory equaled the number of negative eigenvalues in the second variation of the Lagrangian.
Other extremal principles of
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
have been formulated, such as
Gauss' principle of least constraint
The principle of least constraint is one variational formulation of classical mechanics enunciated by Carl Friedrich Gauss in 1829, equivalent to all other formulations of analytical mechanics. Intuitively, it says that the acceleration of a co ...
and its corollary,
Hertz's principle of least curvature
The principle of least constraint is one Variational principle, variational formulation of classical mechanics enunciated by Carl Friedrich Gauss in 1829, equivalent to all other formulations of analytical mechanics. Intuitively, it says that the ...
.
By field
Electromagnetism
The action for
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
is:
:
In relativity theory
The
Einstein–Hilbert action
The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the act ...
which gives rise to the vacuum
Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
is
:
,
where
is the determinant of a spacetime
Lorentz metric
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which t ...
and
is the
scalar curvature
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry ...
.
Quantum mechanics
* Sum over possible paths, Feynman approach. See
Path integral formulation
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional in ...
* Dirac-Frenkel Variational Principle
Apparent teleology
Although equivalent mathematically, there is an important ''philosophical'' difference between the
differential equations of motion
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
and their
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
counterpart. The differential equations are statements about quantities localized to a single point in space or single moment of time. For example,
Newton's second law
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
states that the ''instantaneous'' force
applied to a mass
produces an acceleration
at the same ''instant''. By contrast, the action principle is not localized to a point; rather, it involves integrals over an interval of time and (for fields) extended region of space. Moreover, in the usual formulation of
classical action principles, the initial and final states of the system are fixed, e.g.,
:''Given that the particle begins at position
at time
and ends at position
at time
, the physical trajectory that connects these two endpoints is an extremum of the action integral.''
In particular, the fixing of the ''final'' state appears to give the action principle a
teleological character which has been controversial historically. This apparent
teleology
Teleology (from and )Partridge, Eric. 1977''Origins: A Short Etymological Dictionary of Modern English'' London: Routledge, p. 4187. or finalityDubray, Charles. 2020 912Teleology" In ''The Catholic Encyclopedia'' 14. New York: Robert Appleton ...
is eliminated in the
quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
version of the action principle.
References
* P.L.N. de Maupertuis, ''Accord de différentes lois de la nature qui avaient jusqu'ici paru incompatibles.'' (1744) Mém. As. Sc. Paris p. 417.
* P.L.N. de Maupertuis, ''Le lois de mouvement et du repos, déduites d'un principe de métaphysique.'' (1746) Mém. Ac. Berlin, p. 267.
* 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
scanned copy of complete textat
The Euler Archive', Dartmouth.
* W.R. Hamilton, "On a General Method in Dynamics.", ''Philosophical Transactions of the Royal Society'
Part I (1834) p.247-308Part II (1835) p. 95-144 (''From the collectio
Sir William Rowan Hamilton (1805-1865): Mathematical Papersedited by David R. Wilkins, School of Mathematics, Trinity College, Dublin 2, Ireland. (2000); also reviewed a
On a General Method in Dynamics')
* 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 onlin
Œuvres complètes volume 8a
Gallica-Mathfrom th
Gallica Bibliothèque nationale de France
* Gerhardt CI. (1898) "Über die vier Briefe von Leibniz, die Samuel König in dem Appel au public, Leide MDCCLIII, veröffentlicht hat", ''Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften'', I, 419–427.
* Kabitz W. (1913) "Über eine in Gotha aufgefundene Abschrift des von S. König in seinem Streite mit Maupertuis und der Akademie veröffentlichten, seinerzeit für unecht erklärten Leibnizbriefes", ''Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften'', II, 632–638.
* Marston Morse (1934). "The Calculus of Variations in the Large", ''American Mathematical Society Colloquium Publication'' 18; New York.
* Chris Davis.
' (1998)
* Euler, ''Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes
Additamentum II', Ibid.
* J J O'Connor and E F Robertson,
, (2003), at
The MacTutor History of Mathematics archive'.
* Cassel, Kevin W.: Variational Methods with Applications in Science and Engineering, Cambridge University Press, 2013.
{{DEFAULTSORT:History of Variational Principles in Physics
Calculus of variations
Variational principles in physics