The stationary-action principle – also known as the principle of least action – is a
variational principle
In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those func ...
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 (Ver ...
for that system. The principle states that the trajectories (i.e. the solutions of the equations of motion) are ''
stationary points
In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" in ...
'' 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
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
and
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified 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 (Ver ...
, 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 physics ...
(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 act ...
). 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
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
demonstrated how this principle can be used in quantum calculations by discerning the quantum mechanical underpinning of the principle in the
quantum interference
In physics, interference is a phenomenon in which two waves combine by adding their displacement together at every single point in space and time, to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive ...
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 superflu ...
independently applied this principle in quantum electrodynamics.
The principle remains central in
modern physics
Modern physics is a branch of physics that developed in the early 20th century and onward or branches greatly influenced by early 20th century physics. Notable branches of modern physics include quantum mechanics, special relativity and general ...
and
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 in modern 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 of the ...
,
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 in ...
,
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
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
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 differentiabl ...
.
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, ultraviole ...
. 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 ...
wrote 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 ' ...
.
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.
Scholars often credit
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 ...
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 ...
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 mathem ...
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 , of a physical system is defined as the
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 ...
of the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
''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, to ...
and – technically a
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
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. 3 ...
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:
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,
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 re ...
) 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:
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 obj ...
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 ...
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
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 ...
.
Maupertuis
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 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 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''. 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 cas ...
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
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
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 C ...
entered the debate, defending Maupertuis (the head of his Academy), 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
CI or Ci may refer to:
Business terminology
* Customer intelligence, a discipline in marketing
* Competitive intelligence
* Corporate identity
* Continual improvement
* Confidential information
Businesses and organisations Academia and education ...
in 1898 and
W. Kabitz W. may refer to:
* SoHo (Australian TV channel) (previously W.), an Australian pay television channel
* ''W.'' (film), a 2008 American biographical drama film based on the life of George W. Bush
* "W.", the fifth track from Codeine's 1992 EP ''Bar ...
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 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 potentia ...
, 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
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiaequations 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 ...
of a mechanical body.
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 ...
in 1834 and 1835 applied the variational principle to the classical
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
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 always found minima as opposed to other
stationary points
In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" in ...
(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 functi ...
); 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
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,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 differentiabl ...
. 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 viewpoi ...
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 ...
have been formulated, such as
Gauss's 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 c ...
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 ...
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 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
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 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
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 ...
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
In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
formalism of
path integration
Path integration is the method thought to be used by animals for dead reckoning.
History
Charles Darwin first postulated an inertially-based navigation system in animals in 1873.
The short story ''
Story of Your Life
"Story of Your Life" is a science fiction novella by American writer Ted Chiang, first published in '' Starlight 2'' in 1998, and in 2002 in Chiang's collection of short stories, '' Stories of Your Life and Others''. Its major themes are langu ...
'' 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 the ...
contains visual depictions of
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 ...
along with a discussion of its teleological dimension.
Keith Devlin
Keith J. Devlin (born 16 March 1947) is a British mathematician and popular science writer. Since 1987 he has lived in the United States. He has dual British-American citizenship.
'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)
In physics, action is a scalar quantity describing how a physical system has dynamics (physics), changed over time. Action is significant because the equations of motion of the system can be derived through the principle of stationary action.
I ...
*
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 ...
*
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 acti ...
*
Path of least resistance
The path of least resistance is the physical or metaphorical pathway that provides the least resistance to forward motion by a given object or entity, among a set of alternative paths. The concept is often used to describe why an object or entit ...
*
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
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
*
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-Lou ...
*
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 ...