In
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
, the
principle of least action or more accurately
principle of stationary action is a
variational principleA variational principle is a principle in physics whichis expressed in terms of the calculus of variations.According to Cornelius Lanczos, any physical law which can be expressed as a variational principle describes an expression which is self-adjoint. These expressions are also called Hermitian...
which, when applied to the
actionIn physics, action is an attribute of the development of a physical system. It is a functional which takes the trajectory of the system as its argument and returns a real number as the result....
of a
mechanicalMechanics is the branch of physics concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effect of the bodies on their environment....
system, can be used to obtain the equations of motion for that system. The principle led to the development of the
LagrangianLagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by Italian mathematician Joseph-Louis Lagrange in 1788...
and
HamiltonianHamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
formulations of
classical mechanicsIn the fields of physics, classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the set of physical laws governing and mathematically describing the motions of bodies and aggregates of bodies geometrically distributed within a certain...
.
The principle remains central in
modern physicsThe term modern physics refers to the post-Newtonian conception of physics. The term implies that classical descriptions of phenomena are lacking, and that an accurate, "modern", description of reality requires theories to incorporate elements of quantum mechanics or Einsteinian relativity, or both...
and
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, being applied in the
theory of relativityThe theory of relativity, or simply relativity, generally refers specifically to two theories of Albert Einstein: special relativity and general relativity...
,
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
and
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
, and a focus of modern mathematical investigation in
Morse theoryIn differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...
.
In
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
, the
principle of least action or more accurately
principle of stationary action is a
variational principleA variational principle is a principle in physics whichis expressed in terms of the calculus of variations.According to Cornelius Lanczos, any physical law which can be expressed as a variational principle describes an expression which is self-adjoint. These expressions are also called Hermitian...
which, when applied to the
actionIn physics, action is an attribute of the development of a physical system. It is a functional which takes the trajectory of the system as its argument and returns a real number as the result....
of a
mechanicalMechanics is the branch of physics concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effect of the bodies on their environment....
system, can be used to obtain the equations of motion for that system. The principle led to the development of the
LagrangianLagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by Italian mathematician Joseph-Louis Lagrange in 1788...
and
HamiltonianHamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
formulations of
classical mechanicsIn the fields of physics, classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the set of physical laws governing and mathematically describing the motions of bodies and aggregates of bodies geometrically distributed within a certain...
.
The principle remains central in
modern physicsThe term modern physics refers to the post-Newtonian conception of physics. The term implies that classical descriptions of phenomena are lacking, and that an accurate, "modern", description of reality requires theories to incorporate elements of quantum mechanics or Einsteinian relativity, or both...
and
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, being applied in the
theory of relativityThe theory of relativity, or simply relativity, generally refers specifically to two theories of Albert Einstein: special relativity and general relativity...
,
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
and
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
, and a focus of modern mathematical investigation in
Morse theoryIn differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...
. This article deals primarily with the historical development of the idea; a treatment of the mathematical description and derivation can be found in the article on the
actionIn physics, action is an attribute of the development of a physical system. It is a functional which takes the trajectory of the system as its argument and returns a real number as the result....
. The chief examples of the principle of stationary action are
Maupertuis' principleIn classical mechanics, Maupertuis' principle is an integral equation that determines the path followed by a physical system without specifying the time parameterization of that path. It is a special case of the more generally stated principle of least action...
and
Hamilton's principleIn physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action...
.
The action principle is preceded by earlier ideas in
surveyingSurveying or land surveying is the technique and science of accurately determining the terrestrial or three-dimensional space position of points and the distances and angles between them...
and
opticsOptics is the branch of physics which studies the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
. The rope stretchers of
ancient EgyptAncient Egypt was an ancient civilization of eastern North Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. The civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh, and...
stretched corded ropes between two points to measure distance which minimized the path of separation and Claudius Ptolemy, in his
GeographiaThe Geography is Ptolemy's main work besides the Almagest. It is a treatise on cartography and a compilation of what was known about the world's geography in the Roman Empire of the 2nd century...
(Bk 1, Ch 2), emphasized that one must correct for "deviations from a straight course." In
ancient GreeceAncient Greece is the civilisation belonging to the period of Greek history lasting from the Greek Dark Ages ca. 1100 BC and the Dorian invasion, to 146 BC and the Roman conquest of Greece after the Battle of Corinth. It is generally considered to be the seminal culture which provided the...
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician and is often referred to as the "Father of Geometry." He was active in Hellenistic Alexandria during the reign of Ptolemy I...
states in his
Catoptrica that for the path of light reflecting from a mirror the angle of incidence equals the angle of reflection and
Hero of AlexandriaHero of Alexandria . was an ancient Greek mathematician who was a resident of a Roman province ; he was also an engineer who was active in his native city of Alexandria...
later showed that this path was the shortest length and least time. The credit for the formulation of the principle as it applies to the action is often given to
Pierre-Louis Moreau de MaupertuisPierre-Louis Moreau de Maupertuis was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Berlin Academy of Science, at the invitation of Frederick the Great.Maupertuis made an expedition to Lapland to...
, who wrote about it in 1744 and 1746. However, scholarship indicates that this claim of priority is not so clear;
Leonhard EulerLeonhard Paul Euler was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. His surname is in English ; the common English pronunciation is incorrect....
discussed the principle in 1744, and there is evidence that
Gottfried LeibnizGottfried Wilhelm Leibniz was a German philosopher, polymath and mathematician who wrote primarily in Latin and French....
preceded both by 39 years.
Origins, statement, and the controversy
In the 17th century
Pierre de FermatPierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to modern calculus...
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 principleIn optics, Fermat's principle or the principle of least time is the principle that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light...
.
Credit for the formulation of the
principle of least action is commonly given to
Pierre Louis MaupertuisPierre-Louis Moreau de Maupertuis was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Berlin Academy of Science, at the invitation of Frederick the Great.Maupertuis made an expedition to Lapland to...
, 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". 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 vivaIn the history of science, vis viva is an obsolete scientific theory that served as an elementary and limited early formulation of the principle of conservation of energy...
", twice what we now call the kinetic energy of the system.
Euler's formulation
Leonhard EulerLeonhard Paul Euler was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. His surname is in English ; the common English pronunciation is incorrect....
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 :
As Euler states, is the integral of the momentum over distance traveled 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. Curiously, Euler did not claim any priority, as the following episode shows.
Maupertuis' priority was disputed in 1751 by the mathematician Samuel König, who claimed that it had been invented by
Gottfried LeibnizGottfried Wilhelm Leibniz was a German philosopher, polymath and mathematician who wrote primarily in Latin and French....
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
VoltaireFrançois-Marie Arouet , better known by the pen name Voltaire, was a French Enlightenment writer, essayist, and philosopher known for his wit and his defense of civil liberties, including both freedom of religion and free trade.Voltaire was a prolific writer and produced works in almost every...
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
BernoulliThe Bernoullis were a family of traders and scholars from Basel, Switzerland. The founder of the family, Leon Bernoulli, immigrated to Basel from Antwerp in Flanders in the 16th century, fleeing Spanish oppression....
archives.
Further development
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 energyPotential energy is energy stored within a physical system as a result of the position or configuration of the different parts of that system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do work in the process...
, 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.
Much of the calculus of variations was stated by
Joseph Louis LagrangeJoseph-Louis Lagrange, born Giuseppe Lodovico Lagrangia was an Italian-born mathematician and astronomer, who lived most of his life in Prussia and France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
in 1760 and he proceeded to apply this to problems in dynamics. In
Méchanique Analytique (1788) Lagrange derived the general equations of motion of a mechanical body.
William Rowan HamiltonSir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...
in 1834 and 1835 applied the variational principle to the function to obtain the Lagrangian equations in its present form.
In 1842, Carl Gustav Jacobi tackled the problem of whether the variational principle found minima or other extrema (e.g. a
saddle pointIn mathematics, a saddle point is a point in the domain of a function of two variables which is a stationary point but not a local extremum. At such a point, in general, the surface resembles a saddle that curves up in one direction, and curves down in a different direction...
); most of his work focused on geodesics on two-dimensional surfaces. The first clear general statements were given by
Marston MorseMarston Morse 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 as Morse theory...
in the 1920s and 1930s, leading to what is now known as
Morse theoryIn differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...
. For example, Morse showed that the number of conjugate points in a trajectory equalled the number of negative eigenvalues in the second variation of the Lagrangian.
Other extremal principles of
classical mechanicsIn the fields of physics, classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the set of physical laws governing and mathematically describing the motions of bodies and aggregates of bodies geometrically distributed within a certain...
have been formulated, such as
Gauss' principle of least constraintThe principle of least constraint is another formulation of classical mechanics enunciated by Carl Friedrich Gauss in 1829.The principle of least constraint is a least squares principle stating that the true motion of a mechanical system of masses is the minimum of the quantityfor all trajectories...
and its corollary,
Hertz's principle of least curvatureThe principle of least constraint is another formulation of classical mechanics enunciated by Carl Friedrich Gauss in 1829.The principle of least constraint is a least squares principle stating that the true motion of a mechanical system of masses is the minimum of the quantityfor all trajectories...
.
Apparent teleology
The mathematical equivalence of the
differentialA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
equations of motion and their
integralIn mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...
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 lawNewton's laws of motion are three physical laws that form the basis for classical mechanics. They are:# In the absence of force, a body either is at rest or moves in a straight line with constant speed....
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) an extended region of space. Moreover, in the usual formulation of
classicalWhat "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...
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 characterTeleology is the philosophical study of design and purpose. A teleological school of thought is one that holds all things to be designed for or directed toward a final result, that there is an inherent purpose or final cause for all that exists.As a school of thought it can be contrasted with...
which has been controversial historically. This apparent
teleologyTeleology is the philosophical study of design and purpose. A teleological school of thought is one that holds all things to be designed for or directed toward a final result, that there is an inherent purpose or final cause for all that exists.As a school of thought it can be contrasted with...
is eliminated in the
quantum mechanical versionThe path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...
of the action principle.
See also
- Action (physics)
In physics, action is an attribute of the development of a physical system. It is a functional which takes the trajectory of the system as its argument and returns a real number as the result....
- Calculus of variations
Calculus of variations is a field of mathematics that deals with functionals, as opposed to ordinary calculus which deals with functions. Such functionals can for example be formed as integrals involving an unknown function and its derivatives...
- Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
- Hamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action...
- Lagrangian mechanics
Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by Italian mathematician Joseph-Louis Lagrange in 1788...
- Maupertuis principle
- Path of least resistance
The path of least resistance describes 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 entity takes a given path.In physics, the path of...