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).
[1] 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.
Maupertuis's principle states that the true path of a system described by
is a minimum or a saddle point[2] of the abbreviated action functional,
are the conjugate momenta of the generalized coordinates, defined by the equation
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
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).
For many systems, the kinetic energy
is quadratic in the generalized velocities
may be a complicated function of the generalized coordinates
For such systems, a simple relation relates the kinetic energy, the generalized momenta and the generalized velocities
By defining a normalized distance or metric
The kinetic energy may be written in a massless form
equals the (constant) total energy
In particular, if the potential energy is a constant, then Jacobi's principle reduces to minimizing the path length
in the space of the generalized coordinates, which is equivalent to Hertz's principle of least curvature.
They differ from each other in three important ways: 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.
[3]: 96 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?"
, which was to be minimized over all paths connecting two specified points.
is the velocity of light the corpuscular theory.
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.
[4] Subsequently, the Berlin Academy under Euler's direction declared the letter to be a forgery[5] 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.