Geometrical Derivation of Lagrange’s Equations for a System of Rigid Bodies

Abstract

A geometrical derivation is given for Lagrange’s equations for a system of rigid bodies subject to general holonomic and non-holonomic constraints. As in the case of a similar derivation for a system of particles, the entire system is represented by an abstract particle P moving in a higher-dimensional Euclidean space, called Hertzian space, the metric of which is determined by the radius of gyration of the physical system. The holonomic constraints confine P to move in a Riemannian manifold - the configuration manifold of the constrained system - embedded in Hertzian space. Euler’s laws of linear and angular momenta are expressed as a single balance equation in Hertzian space and Lagrange’s equations emerge as covariant components of this equation taken along the coordinate directions in the configuration manifold. No appeal is made to variational principles or to notions of virtual work.