Maupertuis principle

A principle of least action, the first verbal formulation of which was given by P. Maupertuis. Originally (1744) Maupertuis deduced, from this principle, the laws of reflection and refraction of light, being compatible, in his words, "with the important principle by which nature, when realizing its actions, always goes along the simplest path" (see [1]), and then (1746) he published his universal law of motion and equilibrium: "A general principle: When a change occurs in nature, the quantity of action necessary for this change is least possible. The quantity of action is the product of the masses of the bodies by their speeds and by the distance over which they move" (see [2]). The universality of this principle was justified by Maupertuis with an obscure reasoning of metaphysical nature using teleological arguments, which in the subsequent discussion of his principle gave rise to strong objections by a number of his contemporaries. In addition to the laws of propagation of light, Maupertuis deduced from the principle only the known laws of collisions of bodies and the equilibrium of a lever. In the opinion of J.L. Lagrange "the applications mentioned were of too special a nature for it to be possible to construct from them a proof of a general principle; in addition, they were somewhat indeterminate and arbitrary, which imparts some unreliability to the conclusions which could be made on their basis as well as to the principle itself" (see [3]).

The first mathematical idea of a principle of least action, for the special case of an isolated body, is due to L. Euler. He showed (1744) that under the actions of central forces bodies describe trajectories for which the integral $ \int v d s $ takes a minimum or maximum (see [4]); here $ v $ is the velocity and $ d s $ is the arc element.

For the general case of motion of any system of bodies, interacting in any way so that the total mechanical energy of the system is conserved, the principle of least action was established by Lagrange (1760). Starting from the laws of mechanics he proved that the sum of the products of the masses by the integrals of the velocities multiplied by the elements of the traversed paths, is always a maximum or a minimum (see [5]), that is,

$$ \delta \sum _ { i } m _ {i} \int\limits v _ {i} d s _ {i} = 0 . $$

"This principle, together with the principle of vis viva and developed by the rules of variational calculus, gives immediately all the equations necessary for the solution of each problem; hence there arises a method, as simple as it is general, for solving problems concerning the motions of bodies" (see [3]).

It is accepted, mathematically, to write the principle of least action in Lagrange's form (Lagrange principle), in the form of equality

(see Variational principles of classical mechanics for the equations related to the equation numbers given here). By eliminating the time from

using the energy integral

C.G.J. Jacobi (1837) presented the principle of least action in the form

(see also Jacobi principle).

References

[1]	P. Maupertuis, "Accord de différentes lois de la Nature qui avaient jusqu'ici paru incompatibles" Histore Acad. Sci. Paris (1744)
[2]	P. Maupertuis, "Les lois de mouvement et du répos déduites d'un principle métaphysique" Mem. Acad. R. Sci. et Belles Lettres Berlin (1746) pp. 267–294
[3]	J.L. Lagrange, "Mécanique analytique" , 1–2 , Paris (1788) ((Also: Oeuvres, Vol. 11.))
[4]	L. Euler, "Methodus inveniendi lineas curvas maximi minimive proprietate gandentes sive solutio problematis isoperimetrici latissimo sensu accepti" , Lausanne-Geneva (1744)
[5]	J.L. Lagrange, "Essai d'une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indéfinies" J.-A. Serret (ed.) , Oeuvres , 1 , G. Olms, reprint (1973) pp. 335–362
[6]	C. Jacobi, "Note sur l'intégration des équations différentielles de la dynamique" C.R. Acad. Sci. Paris , 5 (1837) pp. 61–67

References

[a1]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[a2]	F.R. [F.R. Gantmakher] Gantmacher, "Lectures in analytical mechanics" , MIR (1975) (Translated from Russian)
[a3]	E.T. Whittaker, "Analytical dynamics of particles and rigid bodies" , Dover, reprint (1944)