The origins of analytic mechanics in the 18th century

Abstract

Up to the 1740s, problems of equilibrium and motion of material systems were generally solved by an appeal to Newtonian methods for the analysis of forces. Even though, from the very beginning of the century—thanks mainly to Varignon (on which cf. [Blay 1992]), Jean Bernoulli, Hermann and Eu-ler—these methods used the analytical language of the differential calculus, and were considerably improved and simplified, they maintained their essen-tial feature. They were founded on the consideration of a geometric diagram representing the mechanical system under examination, and consequently applied only to (simple) particular and explicitly defined systems. The pos-sibility of expressing the conditions of equilibrium and motion of a general system of bodies, independently of its particular character, only arose when new and essentially non-Newtonian principles were advanced and employed. These principles—such as those of least action and of virtual velocities— are generally known as “variational principles”. In the second half of 18th century, mainly thanks to Lagrange, it was shown that these principles per-mitted the conditions of equilibrium and motion of a mechanical system to be expressed by a suitable system of equations. Though these equations differed when applied to different systems, their form was always the same, whatever the system they were concerned with. As long as this form could be expressed by some general equations, any problem of equilibrium and motion for a particular system could be interpreted as a particular case of one or the other of two general problems: the problem of equilibrium of a mechanical system, and the problem of motion of a mechanical system. Solving these 1 ha ls h