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.
Mathematical formulation.
Maupertuis's principle states that the true path of a system described by formula_1 generalized coordinates formula_2 between two specified states formula_3 and formula_4 is a minimum or a saddle point of the abbreviated action functional,
formula_5
where formula_6 are the conjugate momenta of the generalized coordinates, defined by the equation 
formula_7
where formula_8 is the Lagrangian function for the system. In other words, any "first-order" perturbation of the path results in (at most) "second-order" changes in formula_9. Note that the abbreviated action formula_9 is a functional (i.e. a function from a vector space into its underlying scalar field), which in this case takes as its input a function (i.e. the paths between the two specified states).
Jacobi's formulation.
For many systems, the kinetic energy formula_11 is quadratic in the generalized velocities formula_12
formula_13
although the mass tensor formula_14 may be a complicated function of the generalized coordinates formula_15. For such systems, a simple relation relates the kinetic energy, the generalized momenta and the generalized velocities
formula_16
provided that the potential energy formula_17 does not involve the generalized velocities. By defining a normalized distance or metric formula_18 in the space of generalized coordinates
formula_19
one may immediately recognize the mass tensor as a metric tensor. The kinetic energy may be written in a massless form
formula_20
or,
formula_21
Therefore, the abbreviated action can be written
formula_22
since the kinetic energy formula_23 equals the (constant) total energy formula_24 minus the potential energy formula_17. In particular, if the potential energy is a constant, then Jacobi's principle reduces to minimizing the path length formula_26 in the space of the generalized coordinates, which is equivalent to Hertz's principle of least curvature.
Comparison with Hamilton's principle.
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: 
History.
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 formula_41, which was to be minimized over all paths connecting two specified points. Here formula_42 is the velocity of light the corpuscular theory. Fermat had minimized formula_43 where formula_42 is wave velocity; the two velocities are reciprocal so the two forms are equivalent.
Koenig's claim.
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.
References.
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