Time-integral theorems for nonholonomic systems

Abstract

This paper derives all the important variational and energetic time-integral theorems of linear and nonlinear nonholonomic (and nonconservative) systems, in both real and quasi-generalized coordinates, starting from a simple unifying viewpoint. Specifically, by multiplying the equations of motion with an arbitrary set of functions and then integrating the result between two arbitrary times one arrives at the “generalized virial theorem.” Identifying these arbitrary functions with actual generalized coordinates and velocities, and with virtual and skew (or noncontemporaneous) generalized coordinate variations, one obtains successively the theorems of virial (originally by Clausius), power, and Hamilton's “laws of varying action” for general nonconservative and nonholonomic systems. These theorems can prove useful not only for theoretical arguments and derivations, but also for approximate calculations of oscillating nonholonomic systems. As an illustration Hamilton's principle in quasi-coordinates is used to derive the equations of motion of the famous knife edge/sleigh problem.