Euler's Variational Mechanics

Abstract

Although newtonian mechanics is the most familiar formulation of classical mechanics, we have an another one called variational mechanics. Leonhard Euler attempted to solve some mechanical problems using the latter idea and gave two variational-mechanical laws : "law of rest " in statics, and "law of motion " in dynamics. He maintained that these two laws were "in harmony ", namely that the one could be derived from the other. Moreover, they are both based on "principle of least action, " according to which some quantities are minimum in physical phenomena. In effect, "law of rest " asserts that the sum of "efforts " for bodies is minimum in equilibrium, which Euler interprets as the evidence of the intention of nature to "save efforts. " "Law of motion " determines the trajectory of a mass point, in which "action " is minimum; according to Euler, this could be explained by the inertia of bodies. These laws show Euler's faith in his principle, even though he doesn't use the condition of "minimum " in solving concrete problems; therefore, his "principle of least action " is not a mathmatical requirement. His demonstration of the "harmony " is also based on the belief that nature "saves efforts. " This attempt is, however, a failure in a double meaning; we can't prove the "harmony " from the point of modern view, and Euler's proof is inconsistent with his own "principle of least action. " "Euler's variational mechanics " has failed to realize a unified treatment of statics and dynamics, because it was built up on the "principle of least action. "