Euler's ``harmony'' between the principles of ``rest'' and ``least action''. The conceptual making of analytical mechanics (Q1293451)

In a paper of 1751 presented to the Academy of Berlin, Euler claimed to have established harmony between the quantity of a body that is denoted by the French word \textit{effort}, and its kinetic energy (\textit{force vive}). The mechanical principles invoked by Euler in order to prove this harmony, are the principle of rest and the principle of least action, which had been stated some years earlier by Maupertuis. By bringing the two principles into harmony, Euler shows that there is a relation between action and effect, so that the former is a minimum whenever the latter is a maximum, and vice-versa. According to the author, Euler's proof of harmony between Maupertuis' two principles discloses the physical foundations of analytical mechanics.