INTRODUCTION

Inphysics, theprinciple of least action or, more accurately, theprinciple of stationary action is avariational principlethat, when applied to theactionof amechanicalsystem, can be used to obtain theequations of motionfor that system. The principle led to the development of theLagrangianandHamiltonianformulations ofclassical mechanics. The principle remains central inmodern physicsandmathematics, being applied in thetheory of relativity,quantum mechanicsandquantum field theory, and a focus of modern mathematical investigation inMorse theory.Maupertuis' principleandHamilton's principleexemplify the principle of stationary action. The action principle is preceded by earlier ideas insurveyingandoptics.Rope stretchersinancient Egyptstretched corded ropes to measure the distance between two points.Ptolemy, in hisGeography(Bk 1, Ch 2), emphasized that one must correct for "deviations from a straight course".Inancient Greece,Euclidwrote in hisCatoptricathat, for the path of light reflecting from a mirror, theangle of incidenceequals theangle of reflection Hero of Alexandrialater showed that this path was the shortest length and least time. Scholars often creditPierre Louis Maupertuisfor formulating the principle of least action because he wrote about it in 1744 and 1746. However,Leonhard Eulerdiscussed the principle in 1744,and evidence shows thatGottfried Leibnizpreceded both by 39 years. In 1932,Paul Diracdiscerned thequantum mechanical underpinningof the principle in thequantum interferenceof amplitudes: Formacroscopicsystems, the dominant contribution to the apparent path is the classical path (the stationary, action-extremizing one), while any other path is possible in thequantum realm

GENERAL STATEMENTThe starting point is theaction, denoted(calligraphic S), of a physical system. It is defined as theintegralof theLagrangianLbetween two instants oftimet1andt2- technically afunctionalof theNgeneralized coordinatesq= (q1,q2...qN) which define theconfigurationof the system:

where the dot denotes thetime derivative, andtis time.Mathematically the principle is

where (Greek lowercasedelta) means asmallchange. In words this reads: The path taken by the system between times t1and t2is the one for which theactionisstationary (no change)tofirst order.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 theconfiguration space, i.e. the curveq(t), parameterized by time

As the system evolves,qtraces a path throughconfiguration space(only some are shown). The path taken by the system (red) has a stationary action (S= 0) under small changes in the configuration of the system (q).

Origins, Statements, and ControversyIn the 1600s,Pierre de Fermatpostulated that "light travels between two given points along the path of shortest time," which is known as theprinciple of least timeorFermat's principle. MaupertuisCredit for the formulation of theprinciple of least actionis commonly given toPierre Louis Maupertuis, who 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.Pierre Louis MaupertuisThis 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",Maupertuis' principle

which is the integral of twice what we now call thekinetic energyTof the system.Euler[Leonhard Eulergave a formulation of the action principle in 1744, in very recognizable terms, in theAdditamentum 2to hisMethodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes. Beginning with the second paragraph:Let the mass of the projectile beM, and let its speed bevwhile being moved over an infinitesimal distanceds. The body will have a momentumMvthat, when multiplied by the distanceds, will giveMvds, the momentum of the body integrated over the distanceds. 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 thatMis constant along the path,.

Leonhard Euler

As Euler states, Mvdsis the integral of the momentum over distance travelled, which, in modern notation, equals thereduced actionEuler's principle

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 priorityMaupertuis' priority was disputed in 1751 by the mathematicianSamuel Knig, who claimed that it had been invented byGottfried Leibnizin 1707. Although similar to many of Leibniz's arguments, the principle itself has not been documented in Leibniz's works. Knig himself showed acopyof a 1707 letter from Leibniz toJacob Hermannwith the principle, but theoriginalletter has been lost. In contentious proceedings, Knig was accused of forgery,and even theKing of Prussiaentered the debate, defending Maupertuis (the head of his Academy), whileVoltairedefended Knig.Euler, rather than claiming priority, was a staunch defender of Maupertuis, and Euler himself prosecuted Knig for forgery before the Berlin Academy on 13 April 1752. The claims of forgery were re-examined 150 years later, and archival work byC.I. Gerhardtin 1898 andW. Kabitzin 1913uncovered other copies of the letter, and three others cited by Knig, in theBernoulliarchives.FURTHER DEVELOPMENTEuler continued to write on the topic; in hisReflexions sur quelques loix generales de la nature(1748), he called the quantity "effort". His expression corresponds to what we would now callpotential energy, so that his statement of least action in statics is equivalent to the principle that a system of bodies at rest will adopt a configuration that minimizes total potential energy.

Lagrange and HamiltonMuch of the calculus of variations was stated byJoseph-Louis Lagrangein 1760and he proceeded to apply this to problems in dynamics. InMchanique Analytique(1788) Lagrange derived the generalequations of motionof a mechanical body.William Rowan Hamiltonin 1834 and 1835applied the variational principle to the classicalLagrangianfunction

to obtain theEulerLagrange equationsin their present form.Jacobi and MorseIn 1842,Carl Gustav Jacobitackled the problem of whether the variational principle always found minima as opposed to otherstationary points(maxima or stationarysaddle points); most of his work focused ongeodesicson two-dimensional surfaces.The first clear general statements were given byMarston Morsein the 1920s and 1930s leading to what is now known asMorse theory. For example, Morse showed that the number ofconjugate pointsin a trajectory equalled the number of negative eigenvalues in the second variation of the Lagrangian.Gauss and HertzOther extremal principles ofclassical mechanicshave been formulated, such asGauss's principle of least constraintand its corollary,Hertz's principle of least curvature.Apparent teleologyThe mathematical equivalence of thedifferentialequations of motionand theirintegralcounterpart 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

states that theinstantaneousforceFapplied to a massmproduces an accelerationaat the sameinstant. 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 ofclassicalaction principles, the initial and final states of the system are fixed, e.g.,Given that the particle begins at position x1at time t1and ends at position x2at time t2, the physical trajectory that connects these two endpoints is anextremumof the action integral.In particular, the fixing of thefinalstate appears to give the action principle ateleological characterwhich has been controversial historically.However, some critics maintain this apparentteleologyoccurs 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 thequantumformalism ofpath integration, in which stationary paths are obtained as a result of interference of amplitudes along all possible paths.The short storyStory of Your Lifeby the speculative fiction writerTed Chiangcontains visual depictions ofFermat's Principlealong with a discussion of its teleological dimension.Keith Devlin'sThe Math Instinctcontains 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.