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Principle of least action

a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system

The principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system. In relativity, a different action must be minimized or maximized. The principle can be used to derive Newtonian, Lagrangian, Hamiltonian equations of motion, and even General Relativity. It was historically called "least" because its solution requires finding the path that has the least change from nearby paths. Its classical mechanics and electromagnetic expressions are a consequence of quantum mechanics, but the stationary action method helped in the development of quantum mechanics.

Quotes

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  • The first formulation of (part of) mechanics by means of a variational principle... is due to Maupertuis in 1746 in a paper called "Les lois du mouvement et du repos déduites d'un principe métaphysique" (Laws of motion and rest deduced from a metaphysical principle). Maupertuis had first introduced the principle of least action in optics in 1744. ...Through experimentation, he found that this quantity depends on mass, velocity, and distance. He called the product of the three factors "action" and accordingly expressed a "principle of the least quantity of action"...
    •  Anouk Barberousse, "Form One Version to the Other," Rethinking Scientific Change and Theory Comparison: Stabilities, Ruptures, Incommesurabilities? (2008) ed. Léna Soler, H. Sankey, Paul Hoyningen-Huene
  • Of no little importance are Euler's labors in analytical mechanics. ...He worked out the theory of the rotation of a body around a fixed point, established the general equations of motion of a free body, and the general equation of hydrodynamics. He solved an immense number and variety of mechanical problems, which arose in his mind on all occasions. Thus on reading Virgil's lines. "The anchor drops, the rushing keel is staid," he could not help inquiring what would be the ship's motion in such a case. About the same time as Daniel Bernoulli he published the Principle of the Conservation of Areas and defended the principle of "least action," advanced by P. Maupertius. He wrote also on tides and on sound.
  • Let the mass of the projectile be M, and let its speed be v while being moved over an infinitesimal distance ds. The body will have a momentum Mv that, when multiplied by the distance ds, will give Mv ds, the momentum of the body integrated over the distance ds. Now I assert that the curve thus described by the body to be the curve (from among all other curves connecting the same endpoints) that minimizes
     
    or, provided that M is constant along the path,
     
  • After having worked in the theory of light and gravitation, he announced, in 1744, a new minimum principle, the Principle of Least Action, from which he claimed he could deduce the behavior of light and masses in motion. The principle asserts that nature always behaves so as to minimize an integral known technically as action, and amounting to the integral of the product of mass, velocity, and distance traversed by a moving object. From this principle he deduced the Newtonian laws of motion. With sometimes suitable and sometimes questionable interpretation of the quantities involved, Maupertuis managed to show that optical phenomena, too, could be deduced from this principle. Hence, to an extent at least, he succeeded in uniting the optics of the eighteenth century and mechanical phenomena. ...
    Maupertuis advocated his principle for theological reasons. ...He ...proclaimed his principle to be not only a universal law of nature but also the first scientific proof of the existence of God, for it was "so wise a principle as to be worthy only of a Supreme Being.
    • Morris Kline, Mathematics and the Physical World (1959) Ch. 25: From Calculus to Cosmic Planning, p. 438.
  • The minimum principle that unified the knowledge of light, gravitation, and electricity of Hamilton's time no longer suffices to relate these fundamental branches of physics. Within fifty years of its creation, the belief that Hamilton's principle would outlive all other physical laws of physics was shattered. Minimum principles have since been created for separate branches of physics... but these are not only restricted... but seem to be contrived... The hope of revising the principle so that it will achieve the unification... still drives mathematicians. This is the problem to which... Einstein devoted the last years of his life. Stripped of the theological associations, the belief of a minimum principle still activates physical science. ...
    A single minimum principle, a universal law governing all processes in nature, is still the direction in which the search for simplicity is headed, with the price of simplicity now raised from a mastery of differential equations to a mastery of the calculus of variations.
    • Morris Kline, Mathematics and the Physical World (1959) Ch. 25: From Calculus to Cosmic Planning, p. 442.
  • Maupertuis really had no principle, properly speaking, but only a vague formula, which was forced to do duty as the expression of different familiar phenomena not really brought under one conception. ...Maupertuis' performance, though it had been unfavorably criticized by all mathematicians, is, nevertheless, sort of invested with a sort of historical halo. It would seem almost as if something of the pious faith of the church had crept into mechanics. However, the mere endeavor to gain a more extensive view... was not altogether without results. Euler, at least, if not also Gauss, was stimulated by the attempt of Maupertuis.
  • Euler's view is, that the purposes of the phenomena of nature afford as good a basis of explanation as their causes. If this position is taken, it will be presumed a priori that all natural phenomena present a maximum or a minimum. ...in the solution of mechanical problems... it is possible... to find the expression which in all cases is made maximum or minimum. Euler is thus not led astray... and proceeds much more scientifically than Maupertuis. He seeks an expression whose variation put = 0 gives the ordinary equations of mechanics.
    For a single body moving under the action of forces Euler finds the requisite expression in the formula ∫vds, where ds denotes the element of the path and v the corresponding velocity. This expression is smaller for the path actually taken... therefore, by seeking the path that makes ∫vds a minimum, we can also determine the path. ...In the simplest cases Euler's principle is easily verified. ...
    The consideration of the motion of a projectile... will also show that the quantity ∫vds is smaller for the parabola than for any other neighboring curve; smaller, even, than for the straight line... between the same terminal points. ...
    Jacobi pointed out that we cannot assert that ∫vds for the actual motion is a minimum, but simply that the variation of this expression, in its passage to an infinitely adjacent neighboring path, is = 0. ...unquestionably various other integral expressions may be devised that give by variation the ordinary equations of motion, without its following that the integral expressions in question must possess... any particular physical significance.
    The striking fact remains, however, that so simple and expression as ∫vds does possess the property mentioned.
    • Ernst Mach, The Science of Mechanics (1893) Tr. Thomas J. McCormack
  • I must now explain what I mean by the quantity of action. A certain action is necessary for the carrying of a body from one point to another: this action depends on the velocity which the body has and the space which it describes; but it is neither the velocity nor the space taken separately. The quantity of action varies directly as the velocity and the length of path described; it is proportional to the sum of the spaces, each being multiplied by the velocity with which the body describes it. It is this quantity of action which is here the true expense (dépense) of nature, and which she economizes as much as possible in the motion of light.
  • After so many great men have worked on this subject, I almost do not dare to say that I have discovered the universal principle upon which all these laws are based, a principle that covers both elastic and inelastic collisions and describes the motion and equilibrium of all material bodies.
    This is the principle of least action, a principle so wise and so worthy of the supreme Being, and intrinsic to all natural phenomena; one observes it at work not only in every change, but also in every constancy that Nature exhibits. In the collision of bodies, motion is distributed such that the quantity of action is as small as possible, given that the collision occurs. At equilibrium, the bodies are arranged such that, if they were to undergo a small movement, the quantity of action would be smallest.
    The laws of motion and equilibrium derived from this principle are exactly those observed in Nature. We may admire the applications of this principle in all phenomena: the movement of animals, the growth of plants, the revolutions of the planets, all are consequences of this principle. The spectacle of the universe seems all the more grand and beautiful and worthy of its Author, when one considers that it is all derived from a small number of laws laid down most wisely. Only thus can we gain a fitting idea of the power and wisdom of the supreme Being, not from some small part of creation for which we know neither the construction, usage, nor its relationship to other parts. What satisfaction for the human spirit in contemplating these laws of motion and equilibrium for all bodies in the universe, and in finding within them proof of the existence of Him who governs the universe!
  • When a change occurs in Nature, the quantity of action necessary for that change is as small as possible.
    The quantity of action is the product of the mass of the bodies times their speed and the distance they travel. When a body is transported from one place to another, the action is proportional to the mass of the body, to its speed and to the distance over which it is transported.
  • It [science] has as its highest principle and most coveted aim the solution of the problem to condense all natural phenomena which have been observed and are still to be observed into one simple principle, that allows the computation of past and more especially of future processes from present ones. ...Amid the more or less general laws which mark the achievements of physical science during the course of the last centuries, the principle of least action is perhaps that which, as regards form and content, may claim to come nearest to that ideal final aim of theoretical research.
    • Max Planck, as quoted by Morris Kline, Mathematics and the Physical World (1959) Ch. 25: From Calculus to Cosmic Planning, pp. 441-442.
  • Fermat making use of the argument that Nature could not be wasteful, and was bound for this reason to cause the rays of light to travel between two points in the shortest time possible, was able to deduce from this proposition the laws of reflexion and refraction. Though we do not now attach any weight to the premise, we accept the conclusion.
  • We find... that Mie's Electrodynamics exists in a compressed form in Hamilton's Principle—analogously to the manner in which the development of mechanics attains its zenith in the principle of action. Whereas in mechanics, however, a definite function L of action corresponds to every given mechanical system and has to be deducted from the constitution of the system, we are here concerned with a single system, the world. This is where the real problem of matter takes its beginning: we have to determine the "function of action," the world-function L, belonging to the world. For the present it leaves us in perplexity. If we choose an arbitrary L, we get a "possible" world governed by this function of action, which will be perfectly intelligible to us—more so than the actual world—provided that our mathematical analysis does not fail us. We are, of course, then concerned in discovering the only existing world, the real world for us. Judging from what we know of physical laws, we may expect the L which belongs to it to be distinguished by having simple mathematical properties. Physics, this time as a physics of fields, is again pursuing the object of reducing the totality of natural phenomena to a single physical law: it was believed that this goal was almost within reach once before when Newton's Principia, founded on the physics of mechanical point-masses was celebrating its triumphs. But the treasures of knowledge are not like ripe fruits that may be plucked from a tree.
  • Above all, the ominous clouds of those phenomena that we are with varying success seeking to explain by means of the quantum of action, are throwing their shadows over the sphere of physical knowledge, threatening no one knows what new revolution.
    • Hermann Weyl, Space—Time—Matter (1922) Ch. 3 "Relativity of Space and Time"

The Principle of Least Action (1913)

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Philip Edward Bertrand Jourdain, public domain book @GoogleBooks
  • The present investigations are concerned with the history of the Principle of Least Action in the hands of Maupertuis, Euler, and others. The subject is of great importance in the history of mechanics, both because the principle of least action became, in the hands of Lagrange, "the mother," as Jacobi expressed it, "of our analytical mechanics," and because the animistic tendency displayed in the search for a maximum or a minimum principle in physics undoubtedly had a great influence on such moulders of mechanical theory as Euler, Lagrange (in his early work), Hamilton, Gauss, and in our own times, Willard Gibbs. ...much in this chapter of the evolution of mechanics—one may even say, of thought in general—has been misquoted or misunderstood by even eminent authorities.
  • Besides Lagrange's early printed works, his correspondence with Euler allows us to form some impression of the stimulating effect which the principle of least action had on Lagrange's mind at the beginning of his career. Lagrange's correspondence with Euler extends from 1754... to 1775... Already in 1754 Lagrange announces that he has made "some observations about the maxima and minima which are in the actions of nature." In a letter of August 12, 1755 Lagrange informs Euler that he had a new and simpler method of solving isoperimetrical problems and gives a full statement of it. This discovery of what was afterwards called "the calculus of variations" certainly gave the principle of least action an additional attractiveness to Lagrange; he speaks in a letter of May 19, 1756, of his meditations "on the application of the principle of least action to the whole of dynamics." Lagrange's interest in the principle of least action seems to have evaporated when he observed that, when developed, the integrand is the variational form of d'Alembert's principle, and that it is simpler and equally effective to start with the equations of motion divorced from the integration. This is Lagrange's point of view in 1788. The earliest date at which this change in point of view is... 1764. In a letter of Sept 15, 1782, to Laplace, Lagrange says that he has almost finished a mechanical treatise uniquely founded on "the principle or formula" given in... his memoir of 1780 on the libration of the moon.
  • Maupertuis's first enunciation of the law of the least quantity of action was in a memoir read to the French Academy on April 15th, 1744, entitled "Accord de différentes Loix de la Nature qui avoient jusqu'ici paru incompatibles." The laws in question appear to be those of the reflection and of the refraction of light. When a ray of light in a uniform medium travels from one point to another, either without meeting an obstacle or with meeting a reflecting surface, nature leads it by the shortest path and in the shortest time. But when a ray is refracted by passing from a uniform medium to one of different density, the ray neither describes the shortest space nor does it take the shortest time about it. As Fermat showed, the time would be the shortest if light moved more quickly in rarer media, but Newton proved that, as Descartes had believed, light moves more quickly in denser media. Maupertuis's discovery was that light neither takes always the shortest path nor always that path which it describes in the shortest time, but "that for which the quantity of action is the least."

The Evolution of Scientific Thought from Newton to Einstein (1927)

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A. D'Abro, book @archive.org
  • In order to understand the significance of Action, let us consider any mechanical system passing from an initial configuration P to a final configuration Q. Classical science defined the action A of this system as the difference between its total kinetic energy... and its total potential energy... taken at every instant and then summated over the entire period of time during which the system passed from the initial state P to the final state Q. Now the total kinetic and potential energies of the system at any instant are given by
     
    where T and V represent the densities of the kinetic and potential energies of every point throughout the space occupied by the system. Accordingly, the expression of the action will be given by
     
    ...we have merely replaced (T - V) by a single letter L... referred to as the function of action (also called Lagrangian function).
    Roughly speaking, action was thus in the nature of the product of a duration by an energy contained in a volume of space. On no account may this action be confused with the action dealt with in Newton's law of action and reaction, also expressible as the principle of conservation of momentum. Still less may it be confused with the term "action" which appears in philosophical writings. ...the laws of mechanics can be expressed in a highly condensed form when the concept of action is introduced. Various forms may be given to the principle of Action; here we consider only the form... called Hamilton's Principle of Stationary Action. If we restrict our attention to the very simplest case, we may state Hamilton's principle as follows:
    If we consider all the varied paths along which a conservative system may be guided, so that it will pass in a given time from a definite initial configuration P to a definite configuration Q, we shall find that the course the system actually follows, of its own accord, is always such that along it the action is a minimum (or a maximum).
    ...the principle of action issues ...from the laws of classical mechanics ...A priori, we have no means of deciding whether the laws governing physical phenomena of a non-mechanical nature—those of electromagnetics, for example—would issue from the same principle of action.
  • When Maxwell had proved that his equations of electromagnetics could be thrown into a form compatible with the principle of action, and when he succeeded in amalgamating electricity, magnetism and optics into one science, the universal validity of the principle was accepted. Inasmuch as this principle includes that of the conservation of energy, we can understand why the principle of action was often referred to as the supreme principle of physical science. ...when the principle of action is satisfied by a phenomenon, an indefinite of different mechanical interpretations of the phenomenon are theoretically possible. In the case of electrodynamic phenomena, however, in view of the complicated hypotheses which he was compelled to postulate, Maxwell abandoned all attempts to discover the precise mechanical interpretation which would correspond to reality.
  • The principle... imposes the condition that the natural evolution of any system must be such as to render the action a maximum or a minimum. Could we but express this condition in terms of the usual physical magnitudes, we should be enabled to map out in advance the series of intermediary states through which the phenomenon would pass. From this knowledge we should derive the expression of the laws which governed the evolution of the phenomenon. Here... a twofold problem presents itself. First, we must succeed in finding the correct mathematical expression for the action; and, secondly, we must be in a position to solve the purely mathematical problem of determining under what conditions the action will be a maximum or a minimum.
    Now all problems of maxima and minima are solve by means of the calculus of variations, a form of calculus we owe chiefly to Lagrange. According to the methods of this calculus, we establish under what conditions a magnitude is a maximum or minimum by discovering under what conditions it will be stationary. ...
    When a stone is thrown into the air, it ascends with decreasing speed, then seems to hesitate for a brief period of time as it hovers near the point of maximum height before it starts to fall back again towards the earth. During this brief period of hesitation at the apex of its trajectory, the stone is said to remain "stationary." We can recognize a stationary state by observing that when it is reached no perceptible changes take place over a short period of time. In this way, we understand the connection which exists between the stationary condition and the presence of a maximum or a minimum. In mathematics small variations are represented by the letter δ; hence the stationary condition of the action, or again, the principle of action, is expressed by
     
    ...Lamor applied this method to the phenomena of electricity and magnetism and showed how Maxwell's laws of electrodynamics could be deduced from a suitable mathematical expression L defining the electromagnetic function of action.
  • When the theory of relativity supplanted classical science, it was recognised that the classical equations of mechanics were only approximate, and it became necessary to reformulate the principle of action so as to render it compatible... This work was carried out by the pure mathematicians—by Klein and Hilbert in particular. It was then found that a principle of action differing but slightly from the classical one could be obtained.
  • In classical science, it was strange to find that action... should yet present the artificial aspect of an energy in space multiplied by a duration. As soon, however, as we realise that the fundamental continuum of the universe is one of space-time and not one of separate space and time, the reason for the importance of the seemingly artificial combination of space with time in the expression for the action receives a very simple explanation. Henceforth, action is no longer energy in a volume of space multiplied by a duration; it is simply energy in a volume of the world, that is to say, in a volume of four-dimensional space-time. Designating a volume of space-time by  , we have
     
    so that our principle of action,   becomes
     
    Now there is a perfect symmetry between the rôles of space and time.
  • From the expression for the atom of energy or quantum hv, where h is a constant and v is the frequency of the radiation, it is obvious that there exist as many different types of quanta of energy as there exist different frequencies of radiation. There is no unique type of quantum of energy in nature. That which is universal is not the quantum of energy hv, but the constant h. It can be shown that Planck's constant h is not a mere number; it represents some definite abstract mathematical entity, and that entity is action.
    We must assume, therefore, that there exist atoms of action in nature, just as there exist atoms of matter. ...we must possess a fairly thorough understanding of what is meant by action, as also of the part this important entity plays in science. ...the atomicity of action ...suggests that change is always discontinuous ...a series of jerks or jumps.

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