Introduction

This chapter develops several basic ideas of dynamics, emphasizing general principles that are useful in classifying the behaviour of dynamical systems. The reader is assumed to be familiar with elementary concepts of classical mechanics. Concepts developed here will be needed in the study of special relativity (Chap. 3), statistical mechanics (Chap. 5), general relativity (Chap. 11), Sun and solar system (Vol. II), binary stars (Vol. II), and galactic dynamics (Vol. III).

Time Evolution of Dynamical Systems

Many systems encountered in nature can be described by a finite set of N real variables [q1(t), q2(t), …, qi(t), …, qN(t)] t h a t evolve in time. For example, in the study of two stars, moving under the influence of their mutual gravitational force, we are interested in the positions of the stars as functions of time. The position of each star can be described by three coordinates (in three-dimensional space) so that the full system can be described by a total of six functions of time. The quantities qi(t) (with i = 1,2, …, N) are called dynamical variables; obviously, we are free to choose any other set of N independent, single-valued functions of qi as dynamical variables to describe the system, with the particular choice often dictated by mathematical convenience. The central problem of dynamics is related to determining the time dependence of qi(t) and studying the general characteristics of motion.