Translation talk:Derivation of the laws of motion and equilibrium from a metaphysical principle

Hi, I undertook this translation of Maupertuis' second article on the principle of least action, because I needed it for a planned series of articles about the various types of the "action principle" in physics and this history of variational principles in physics. It was fun, especially seeing how the science of mechanics developed.

The original manuscript from which I worked can be found here. The original was added to the French Wikisource for easier reading.

This translation of Maupertuis' Les loix du mouvement et du repos déduites d'un principe metaphysique is hereby released into the public domain. Use it well. Willow 20:49, 17 July 2006 (UTC)

The original title is


 * Les loix du mouvement et du repos déduites d'un principe metaphysique

Maupertuis originally presented this work as a talk to the Berlin Academy of Sciences on 6 October 1746, under the title


 * Les loix du mouvement et du repos déduites des attributs de Dieu (Derivation of the laws of motion and equilibrium from the attributes of God).

I corrected a typo in the original text of Problem I of section three; see if you can find it! :)

Translator's historical notes:

Maupertuis takes credit here for inventing the general principle of least action first (conveniently labeled in section three as General principle). This seems to be a mistaken impression on Maupertuis' part. The general metaphysical principle that "Nature acts by the most economical means" was well-known in Maupertuis' time, having originated among the ancient Greeks. Maupertuis felt that his contribution was his mathematical definition of the action, which he gives only for light in his 1744 article. The definition of a principle of least action for the motion of material particles was first published by Euler in his Methodus inveniendi/Additamentum II (also published in 1744).

Interestingly, Maupertuis' mathematical definition of the action seems to evolve, along with his conception of how it should be minimized. In his 1744 article, Maupertuis defines the action for light as $$\int v ds$$ and minimizes over all paths between two given points. For comparison, in his Additamentum II, Euler defines the action for a material particle in its modern form, $$\int m v \, ds$$ (or as $$\int v \, ds$$ when $$m$$ is constant) and likewise minimizes over all paths joining two points that completely define the initial and final states. Euler also recognizes that the principle only holds when the speed can be determined from position, i.e., when energy is conserved. Subsequent research has not altered these basic ideas, except to allow the action to be a stationary point (not just a minimum).

In this article, however, Maupertuis gives two different definitions that do not seem to agree with the modern definition:


 * For collisions, Maupertuis defines the action as $$\sum m_{k} \left( \Delta v_{k} \right)^{2}$$ (i.e., the sum over the particles of their mass multiplied by their squared changes in speed) and minimizes with respect to the final speeds of the particles. Additional constraints are needed to relate the final speeds for the minimization, e.g., by requiring that they be the identical (perfectly inelastic collisions, which do not conserve energy) or that the relative speed be unchanged after the collision (perfectly elastic collisions).


 * For mechanical equilibrium (the lever problem), he defines it as $$\sum m_{k} ds_{k}^{2}$$ (i.e., the sum over the particles of their mass multiplied by the square of their infinitesimal distance travelled) and minimizes with the respect to the position of the fulcrum point of the lever.

In all three problems, Maupertuis appears to derive the correct answers (which were already known) by an ad hoc method.