Formulation of the Governing Equations of Motion of Dynamic Systems on a Principal Bundle

Abstract

In this paper, we present a geometric approach to form the governing equations of motion of dynamic systems. A geometric form of the d’Alembert-Lagrange equation on the configuration manifold is first developed and extended to a principal bundle. We can then obtain the explicit form of equations of motion by explicating the geometric form in a coordinate neighborhood on the principal bundle. This approach conveniently permits the choice of quantities to be used which best describe configurations, motions or constraints, and it yields equations of motion in concise forms. Examples are presented to illustrate the use and effectiveness of the approach. The objective of this paper is to provide a general geometric perspective of the governing equations of motion, and explain its suitability for studying complex dynamic systems subject to non-holonomic constraints.