In classical mechanics, Maupertuis' principle (named after Pierre Louis Maupertuis), is 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.
Contents
Mathematical formulation
Maupertuis' principle states that the true path of a system described by
where
where
Jacobi's formulation
For many systems, the kinetic energy
although the mass tensor
provided that the potential energy
one may immediately recognize the mass tensor as a metric tensor. The kinetic energy may be written in a massless form
or, equivalently,
Hence, the abbreviated action can be written
since the kinetic energy
Comparison with Hamilton's principle
Hamilton's principle and Maupertuis' principle are occasionally confused 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, where he defined action as
A few months later, well before Maupertuis' work appeared in print, Leonhard Euler independently defined action in its modern abbreviated form
Two years later, Maupertuis cites Euler's 1744 work as a "beautiful application of my principle to the motion of the planets" and goes on to apply the principle of least action to the lever problem in mechanical equilibrium and to perfectly elastic and perfectly inelastic collisions (see the 1746 publication below). Thus, Maupertuis takes credit for conceiving the principle of least action as a general principle applicable to all physical systems (not merely to light), whereas the historical evidence suggests that Euler was the one to make this intuitive leap. Notably, Maupertuis' definitions of the action and protocols for minimizing it in this paper are inconsistent with the modern approach described above. Thus, Maupertuis' published work does not contain a single example in which he used Maupertuis' principle (as presently understood).
In 1751, Maupertuis' 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 Leibniz that described results similar to those derived by Euler in 1744. However, Maupertuis and others demanded that Koenig produce the original of the letter to authenticate its having been written by Leibniz. Koenig only had a copy and no clue as to the whereabouts of the original. Consequently, the Berlin Academy under Euler's direction declared the letter to be a forgery and that its President, Maupertuis, could continue to claim priority for having invented the principle. Koenig continued to fight for Leibniz's priority and soon luminaries such as Voltaire and the King of Prussia, Frederick II were engaged in the quarrel. However, no progress was made until the turn of the twentieth century, when other independent copies of Leibniz's letter were discovered. The present scholarly consensus seems to be that the quotations from Leibniz are indeed genuine, i.e., that he had invented Maupertuis' principle and applied it to several mechanical problems by 1707 (37 years before Maupertuis and Euler) but did not publish his findings.