Физикада ең аз ұзындық принципі

Principle of least length in physics
In classical mechanics, Maupertuis's principle (named after Pierre Louis Maupertuis, 1698 – 1759) 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 the more generally stated principle of least action. Using the calculus of variations, it results in an integral equation formulation of the equations of motion for the system.

Hamilton's principle and Maupertuis's principle are occasionally confused with each other and both have been called the principle of least action. They differ from each other in three important ways:
their definition of the action
the solution that they determine Hamilton's principle determines the trajectory as a function of time, whereas Maupertuis's principle determines only the shape of the trajectory in the generalized coordinates. For example, Maupertuis's principle determines the shape of the ellipse on which a particle moves under the influence of an inverse square central force such as gravity, but does not describe per se how the particle moves along that trajectory. (However, this time parameterization may be determined from the trajectory itself in subsequent calculations using the conservation of energy.) By contrast, Hamilton's principle directly specifies the motion along the ellipse as a function of time. and the constraints on the variation. Maupertuis's principle requires that the two endpoint states and be given and that energy be conserved along every trajectory. By contrast, Hamilton's principle does not require the conservation of energy, but does require that the endpoint times and be specified as well as the endpoint states and .

Maupertuis was the first to publish a principle of least action, as a way of adapting Fermat's principle for waves to a corpuscular (particle) theory of light. Pierre de Fermat had explained Snell's law for the refraction of light by assuming light follows the path of shortest time, not distance. This troubled Maupertuis, since he felt that time and distance should be on an equal footing: "why should light prefer the path of shortest time over that of distance?" Maupertuis defined his action as , which was to be minimized over all paths connecting two specified points. Here is the velocity of light the corpuscular theory. Fermat had minimized where is wave velocity; the two velocities are reciprocal so the two forms are equivalent.

In 1751, Maupertuis's priority for the principle of least action was challenged in print (Nova Acta Eruditorum of Leipzig) by an old acquaintance, Johann Samuel Koenig, who quoted a 1707 letter purportedly from Gottfried Wilhelm Leibniz to Jakob Hermann that described results similar to those derived by Leonhard Euler in 1744. Maupertuis and others demanded that Koenig produce the original of the letter to authenticate its having been written by Leibniz. Leibniz died in 1716 and Hermann in 1733, so neither could vouch for Koenig. Koenig claimed to have the letter copied from the original owned by Samuel Henzi, and no clue as to the whereabouts of the original, as Henzi had been executed in 1749 for organizing the Henzi conspiracy for overthrowing the aristocratic government of Bern. Subsequently, the Berlin Academy under Euler's direction declared the letter to be a forgery and that Maupertuis, could continue to claim priority for having invented the principle. Curiously Voltaire got involved in the quarrel by composing Diatribe du docteur Akakia ("Diatribe of Doctor Akakia") to satirize Maupertuis' scientific theories (not limited to the principle of least action). While this work damaged Maupertuis's reputation, his claim to priority for least action remains secure.