For the sake of simplicity, and brevity, we shall restrict our investigations to the motions of idealized point particles and idealized rigid bodies. To be more exact, we shall exclude from consideration any discussion of statics, the strength of materials, and the non-rigid motions of continuous media. We shall also concentrate, for the most part, on motions which take place under the influence of conservative forces, such as gravity, which can be accurately represented in terms of simple mathematical formulae. Finally, with one major exception, we shall only consider that subset of dynamical problems that can be solved by means of conventional mathematical analysis.
Newtonian dynamics was originally developed in order to predict the motions of the objects which make up the Solar System. It turns out that this is an ideal application of the theory, since the objects in question can be modeled as being rigid to a fair degree of accuracy, and the motions take place under the action of a single conservative force--namely, gravity--that has a simple mathematical form. In particular, the frictional forces which greatly complicate the application of Newtonian dynamics to the motions of everyday objects close to the Earth's surface are completely absent. Consequently, in this book we shall make a particular effort to describe how Newtonian dynamics can successfully account for a wide variety of different solar system phenomena. For example, during the course of this book, we shall explain the origins of Kepler's laws of planetary motion (see Chapter 5), the rotational flattening of the Earth, the tides, the Roche radius (i.e., the minimum radius at which a moon can orbit a planet without being destroyed by tidal forces), the forced precession and nutation of the Earth's axis of rotation, and the forced perihelion precession of the planets (see Chapter 12). We shall also derive the Tisserand criterion used to re-identify comets whose orbits have been modified by close encounters with massive planets, account for the existence of the so-called Trojan asteroids which share the orbit of Jupiter (see Chapter 13), and analyze the motion of the Moon (see Chapter 14).
Virtually all of the results described in this book were first obtained--either by Newton himself, or by scientists living in the 150, or so, years immediately following the initial publication of his theory--by means of conventional mathematical analysis. Indeed, scientists at the beginning of the 20th century generally assumed that they knew everything that there was to known about Newtonian dynamics. However, they were mistaken. The advent of fast electronic computers, in the latter half of the 20th century, allowed scientists to solve nonlinear equations of motion, for the first time, via numerical techniques. In general, such equations are insoluble using standard analytic methods. The numerical investigation of dynamical systems with nonlinear equations of motion revealed the existence of a previously unknown type of motion known as deterministic chaos. Such motion is quasi-random (despite being derived from deterministic equations of motion), aperiodic, and exhibits extreme sensitivity to initial conditions. The discovery of chaotic motion lead to a renaissance in the study of Newtonian dynamics which started in the late 20th century and is still ongoing. It is therefore appropriate that the last chapter in this book is devoted to an in-depth numerical investigation of a particular dynamical system that exhibits chaotic motion (see Chapter 15).