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:''This article discusses the history of the principle of least action. For the application, please refer to action (physics).'' 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.〔Chapter 19 of Volume II, Feynman R, Leighton R, and Sands M. ''The Feynman Lectures on Physics ''. 3 volumes 1964, 1966. Library of Congress Catalog Card No. 63-20717. ISBN 0-201-02115-3 (1970 paperback three-volume set); ISBN 0-201-50064-7 (1989 commemorative hardcover three-volume set); ISBN 0-8053-9045-6 (2006 the definitive edition (2nd printing); hardcover)〕 Its classical mechanics and electromagnetic expressions are a consequence of quantum mechanics, but the stationary action method helped in the development of quantum mechanics.〔"The Character of Physical Law" Richard Feynman〕 The principle remains central in modern physics and mathematics, being applied in the theory of relativity, quantum mechanics and quantum field theory, and a focus of modern mathematical investigation in Morse theory. Maupertuis' principle and Hamilton's principle exemplify the principle of stationary action. The action principle is preceded by earlier ideas in surveying and optics. Rope stretchers in ancient Egypt stretched corded ropes to measure the distance between two points. Ptolemy, in his ''Geography'' (Bk 1, Ch 2), emphasized that one must correct for "deviations from a straight course". In ancient Greece, Euclid 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 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 1744〔P.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 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 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.〕 In 1932, Paul Dirac discerned the quantum mechanical underpinning of the principle in the quantum interference of amplitudes:For macroscopic systems, the dominant contribution to the apparent path is the classical path (the stationary, action-extremizing one), while any other path is possible in the quantum realm. ==General statement== The starting point is the ''action'', denoted (calligraphic S), of a physical system. It is defined as the integral of the Lagrangian ''L'' between two instants of time ''t''1 and ''t''2 - technically a functional of the ''N'' generalized coordinates q = (''q''1, ''q''2 ... ''qN'') which define the configuration of the system: : where δ (Greek lowercase delta) means a ''small'' change. In words this reads:〔 :''The path taken by the system between times t1 and t2 is the one for which the action is stationary (no change) to first order.'' In applications the statement and definition of action are taken together:〔Classical Mechanics, T.W.B. Kibble, European Physics Series, McGraw-Hill (UK), 1973, ISBN 0-07-084018-0〕 : 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 q(''t''), parameterized by time (see also parametric equation for this concept). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Principle of least action」の詳細全文を読む スポンサード リンク
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