Maupertuis's principle states that the true path of a system described by generalized coordinates between two specified states and is a stationary point of the abbreviated action functional where are the conjugate momenta of the generalized coordinates, defined by the equation where is the Lagrangian function for the system. In other words, any first-order perturbation of the path results in second-order changes in. Note that the abbreviated action is a functional, which in this case takes as its input a function.
Jacobi's formulation
For many systems, the kinetic energy is quadratic in the generalized velocities although the mass tensor 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 provided that the potential energy does not involve the generalized velocities. By defining a normalized distance or metric in the space of generalized coordinates one may immediately recognize the mass tensor as a metric tensor. The kinetic energy may be written in a massless form or, Therefore, the abbreviated action can be written since the kinetic energy equals the total energy minus the potential 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.
and Maupertuis's principle are occasionally confused 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...
...and the constraints on the variation.
History
Maupertuis was the first to publish a principle of least action, where he defined action as, which was to be minimized over all paths connecting two specified points. However, Maupertuis applied the principle only to light, not matter. He arrived at the principle by considering Snell's law for the refraction of light, which Fermat had explained by Fermat's principle, that 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?" Accordingly, Maupertuis asserts with no further justification the principle of least action as equivalent but more fundamental than Fermat's principle, and uses it to derive Snell's law. Maupertuis specifically states that light does not follow the same laws as material objects. A few months later, well before Maupertuis's work appeared in print, Leonhard Euler independently defined action in its modern abbreviated form and applied it to the motion of a particle, but not to light. Euler also recognized that the principle only held when the speed was a function only of position, i.e., when the total energy was conserved. Euler used this principle to derive the equations of motion of a particle in uniform motion, in a uniform and non-uniform force field, and in a central force field. Euler's approach is entirely consistent with the modern understanding of Maupertuis's principle described above, except that he insisted that the action should always be a minimum, rather than a stationary point. 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. Thus, Maupertuis takes credit for conceiving the principle of least action as a general principle applicable to all physical systems, whereas the historical evidence suggests that Euler was the one to make this intuitive leap. Notably, Maupertuis's definitions of the action and protocols for minimizing it in this paper are inconsistent with the modern approach described above. Thus, Maupertuis's published work does not contain a single example in which he used Maupertuis's principle. In 1751, Maupertuis's priority for the principle of least action was challenged in print 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, 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.
