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The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the ''
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
'' of a
mechanical Mechanical may refer to: Machine * Machine (mechanical), a system of mechanisms that shape the actuator input to achieve a specific application of output forces and movement * Mechanical calculator, a device used to perform the basic operations of ...
system, yields the
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 (Verla ...
for that system. The principle states that the trajectories (i.e. the solutions of the equations of motion) are '' stationary points'' of the system's ''action functional''. The term "least action" is a historical misnomer since the principle has no minimality requirement: the value of the action functional need not be minimal (even locally) on the trajectories.The Feynman Lectures on Physics Vol. II Ch. 19: The Principle of Least Action
/ref> The principle can be used to derive Newtonian, Lagrangian and Hamiltonian
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 (Verla ...
, and even
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physic ...
(see
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 ac ...
). In relativity, a different action must be minimized or maximized. The classical mechanics and electromagnetic expressions are a consequence of quantum mechanics. The stationary action method helped in the development of quantum mechanics. In 1933, the physicist Paul Dirac demonstrated how this principle can be used in quantum calculations by discerning the quantum mechanical underpinning of the principle in the quantum interference of amplitudes. Subsequently
Julian Schwinger Julian Seymour Schwinger (; February 12, 1918 – July 16, 1994) was a Nobel Prize winning American theoretical physicist. He is best known for his work on quantum electrodynamics (QED), in particular for developing a relativistically invariant ...
and
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfl ...
independently applied this principle in quantum electrodynamics. The principle remains central in modern physics and mathematics, being applied in
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws o ...
,
fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
, the
theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena i ...
,
quantum mechanics 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, ...
,
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, and string theory and is a focus of modern mathematical investigation in
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 differentia ...
.
Maupertuis' principle In classical mechanics, Maupertuis's principle (named after Pierre Louis Maupertuis) states that the path followed by a physical system is the one of least length (with a suitable interpretation of ''path'' and ''length''). It is a special case of ...
and
Hamilton's principle In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function ...
exemplify the principle of stationary action. The action principle is preceded by earlier ideas in
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, ultravio ...
. 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, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ge ...
wrote in his ''Catoptrica'' that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection.
Hero of Alexandria Hero of Alexandria (; grc-gre, Ἥρων ὁ Ἀλεξανδρεύς, ''Heron ho Alexandreus'', also known as Heron of Alexandria ; 60 AD) was a Greek mathematician and engineer who was active in his native city of Alexandria, Roman Egypt. He ...
later showed that this path was the shortest length and least time. Scholars often credit Pierre Louis Maupertuis for formulating the principle of least action because he wrote about it in 1744P.L.M. de Maupertuis, '' Accord de différentes lois de la nature qui avaient jusqu'ici paru incompatibles.'' (1744) Mém. As. Sc. Paris p. 417. ( English translation) and 1746.P.L.M. de Maupertuis, '' Le lois de mouvement et du repos, déduites d'un principe de métaphysique.'' (1746) Mém. Ac. Berlin, p. 267.( English translation) However,
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 ...
discussed the principle in 1744,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 text
at
The Euler Archive
', Dartmouth.
and evidence shows that
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 mat ...
preceded both by 39 years.J J O'Connor and E F Robertson,
The Berlin Academy and forgery
, (2003), at
The MacTutor History of Mathematics archive
'.
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.


General statement

The ''
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
'', denoted \mathcal , of a physical system is defined as the
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of the Lagrangian ''L'' between two instants of
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...
and – technically a functional of the
generalized coordinates In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
which are functions of time and define the
configuration Configuration or configurations may refer to: Computing * Computer configuration or system configuration * Configuration file, a software file used to configure the initial settings for a computer program * Configurator, also known as choice board ...
of the system: \mathbf : \mathbf \to \mathbf^N \mathcal mathbf, t_1, t_2= \int_^ L(\mathbf(t),\mathbf(t), t) dt where the dot denotes the
time derivative A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t. Notation A variety of notations are used to denote th ...
, and is time. Mathematically the principle isAnalytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, \delta \mathcal = 0 , where ''δ'' (lowercase Greek
delta Delta commonly refers to: * Delta (letter) (Δ or δ), a letter of the Greek alphabet * River delta, at a river mouth * D ( NATO phonetic alphabet: "Delta") * Delta Air Lines, US * Delta variant of SARS-CoV-2 that causes COVID-19 Delta may also ...
) means a ''small'' change. In words this reads: Stationary action is not always a minimum, despite the historical name of least action. It is a minimum principle for sufficiently short, finite segments in the path. In applications the statement and definition of action are taken together: \delta \int_^ L(\mathbf, \mathbf,t) dt = 0 . The action and Lagrangian both contain the dynamics of the system for all times. The term "path" simply refers to a curve traced out by the system in terms of the coordinates in the configuration space, i.e. the curve , parameterized by time (see also
parametric equation In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric ob ...
for this concept).


Origins, statements, and controversy


Fermat

In the 1600s,
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 i ...
postulated that "''light travels between two given points along the path of shortest time''," which is known as the principle of least time or Fermat's principle.


Maupertuis

Credit for the formulation of the principle of least action is commonly given to Pierre Louis Maupertuis, who felt that "Nature is thrifty in all its actions", and applied the principle broadly: This notion of Maupertuis, although somewhat deterministic today, does capture much of the essence of 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 "
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 ...
", which is the integral of twice what we now call the
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
''T'' of the system.


Euler

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 ...
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''. Beginning with the second paragraph: As Euler states, is the integral of the momentum over distance travelled, which, in modern notation, equals the abbreviated or
reduced action In physics, action is a scalar quantity describing how a physical system has changed over time. Action is significant because the equations of motion of the system can be derived through the principle of stationary action. In the simple case ...
Thus, Euler made an equivalent and (apparently) independent statement of the variational principle in the same year as Maupertuis, albeit slightly later. Curiously, Euler did not claim any priority, as the following episode shows.


Disputed priority

Maupertuis' priority was disputed in 1751 by the mathematician Samuel König, 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 mat ...
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 The monarchs of Prussia were members of the House of Hohenzollern who were the hereditary rulers of the former German state of Prussia from its founding in 1525 as the Duchy of Prussia. The Duchy had evolved out of the Teutonic Order, a Roman ...
entered the debate, defending Maupertuis (the head of his Academy), while
Voltaire François-Marie Arouet (; 21 November 169430 May 1778) was a French Enlightenment writer, historian, and philosopher. Known by his '' nom de plume'' M. de Voltaire (; also ; ), he was famous for his wit, and his criticism of Christianity— ...
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 archives.


Further development

Euler continued to write on the topic; in his ''Réflexions sur quelques loix générales de la nature'' (1748), he called action "effort". His expression corresponds to modern
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 potent ...
, and his statement of least action says that the total potential energy of a system of bodies at rest is minimized, a principle of modern statics.


Lagrange and Hamilton

Much of the calculus of variations was stated by Joseph-Louis Lagrange in 1760 and he proceeded to apply this to problems in dynamics. In ''Mécanique analytique'' (1788) Lagrange derived the general
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 (Verla ...
of a mechanical body.
William Rowan Hamilton Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Irel ...
in 1834 and 1835 applied the variational principle to the classical Lagrangian function L = T - V to obtain the Euler–Lagrange equations in their present form.


Jacobi, Morse and Caratheodory

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, differential equations, determinants, and number theory. His name is occasional ...
tackled the problem of whether the variational principle always found minima as opposed to other stationary points (maxima or stationary
saddle points 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 In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
on two-dimensional surfaces. The first clear general statements were given by Marston Morse in the 1920s and 1930s,Marston Morse (1934). "The Calculus of Variations in the Large", ''American Mathematical Society Colloquium Publication'' 18; New York. 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 differentia ...
. For example, Morse showed that the number of
conjugate points In differential geometry, conjugate points or focal points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian. Another viewpoin ...
in a trajectory equalled the number of negative eigenvalues in the second variation of the Lagrangian. A particularly elegant derivation of the Euler-Lagrange equation was formulated by Constantin Caratheodory and published by him in 1935.


Gauss and Hertz

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 m ...
have been formulated, such as Gauss's principle of least constraint and its corollary, Hertz's principle of least curvature.


Disputes about possible teleological aspects

The mathematical equivalence of 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 (Verla ...
and their
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
counterpart has important philosophical implications. 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 \mathbf = m\mathbf states that the ''instantaneous'' force F applied to a mass ''m'' produces an acceleration a 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) an 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., In particular, the fixing of the ''final'' state has been interpreted as giving the action principle a teleological character which has been controversial historically. However, according to W. Yourgrau and S. Mandelstam, ''the teleological approach... presupposes that the variational principles themselves have mathematical characteristics which they ''de facto'' do not possess'' In addition, some critics maintain this apparent teleology occurs because of the way in which the question was asked. By specifying some but not all aspects of both the initial and final conditions (the positions but not the velocities) we are making some inferences about the initial conditions from the final conditions, and it is this "backward" inference that can be seen as a teleological explanation. Teleology can also be overcome if we consider the classical description as a limiting case of the quantum formalism of path integration, in which stationary paths are obtained as a result of interference of amplitudes along all possible paths. The short story '' Story of Your Life'' by the speculative fiction writer
Ted Chiang Ted Chiang (born 1967) is an American science fiction writer. His work has won four Nebula awards, four Hugo awards, the John W. Campbell Award for Best New Writer, and six Locus awards. His short story " Story of Your Life" was the basis of th ...
contains visual depictions of Fermat's Principle along with a discussion of its teleological dimension. Keith Devlin's ''The Math Instinct'' contains a chapter, "Elvis the Welsh Corgi Who Can Do Calculus" that discusses the calculus "embedded" in some animals as they solve the "least time" problem in actual situations.


See also

* Action (physics) *
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 ...
*
Schwinger's quantum action principle The Schwinger's quantum action principle is a variational approach to quantum mechanics and quantum field theory. This theory was introduced by Julian Schwinger in a series of articles starting 1950. Approach In Schwingers approach, the a ...
* Path of least resistance *
Analytical mechanics In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the ...
*
Calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
* Hamiltonian mechanics *
Lagrangian mechanics 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-Lo ...
*
Occam's razor Occam's razor, Ockham's razor, or Ocham's razor ( la, novacula Occami), also known as the principle of parsimony or the law of parsimony ( la, lex parsimoniae), is the problem-solving principle that "entities should not be multiplied beyond neces ...


Notes and references


External links


Interactive explanation of the principle of least action

Interactive applet to construct trajectories using principle of least action
* * *

{{DEFAULTSORT:Principle Of Least Action Concepts in physics Variational principles History of physics Scientific laws de:Prinzip der kleinsten Wirkung sq:Principi i Hamiltonit