External obstacles: Lagrange and the mechanics of constraints

Abstract

Abstract The zenith of theory-building in the century after the Principia was Lagrange’s treatise on “analytic” mechanics. In this chapter, the authors address two themes: his results in the rational mechanics of constrained motions, and their relevance for philosophical mechanics. The problem of external constraints found a wholesale solution in Lagrange’s analytic mechanics of 1788. Two ingredients were key to his solution: a dynamical law, viz. the Principle of Virtual Velocities; and the method of Lagrange multipliers. This combination allowed him to unify all rational mechanics then available. The authors assess Lagrange’s achievement, via a constructive and then a principle reading of his theory.