Physics and Geometry

Abstract

This chapter emphasizes the importance of a geometric approach to dynamics. The central objects of interest are trajectories of a dynamical system through multidimensional spaces composed of generalized coordinates. Trajectories through configuration space are parameterized by the path length element, which becomes an important feature in later chapters on relativity and metric spaces. Trajectories through state space are defined by mathematical flow equations whose flow fields and flow lines become the chief visualization tool for complex dynamics. Coordinate transformations and Jacobian matrices are used throughout this text, and the transformation to noninertial frames introduces fictitious forces like the Coriolis force that are experienced by observers in noninertial frames. Uniformly rotating frames provide the noninertial reference frames for the description of rigid-body motion.