Introduction

Newtonian mechanics is a mathematical model whose purpose is to account for the motions of the various objects in the universe. The general principles of this model were first enunciated by Sir Isaac Newton in a work entitled Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). This work, which was published in 1687, is nowadays more commonly referred to as the Principia.^2.1

Until the beginning of the twentieth century, Newtonian mechanics was thought to constitute a complete description of all types of motion occurring in the universe. We now know that this is not the case. The modern view is that Newton's model is only an approximation that is valid under certain circumstances. The model breaks down when the velocities of the objects under investigation approach the speed of light in a vacuum, and must be modified in accordance with Einstein's special theory of relativity. The model also fails in regions of space that are sufficiently curved that the propositions of Euclidean geometry do not hold to a good approximation, and must be augmented by Einstein's general theory of relativity. Finally, the model breaks down on atomic and subatomic lengthscales, and must be replaced by quantum mechanics. In this book, we shall (almost entirely) neglect relativistic and quantum effects. It follows that we must restrict our investigations to the motions of large (compared to an atom), slow (compared to the speed of light) objects moving in Euclidean space. Fortunately, virtually all of the motions encountered in conventional celestial mechanics fall into this category.

Newton very deliberately modeled his approach in the Principia on that taken in Euclid's Elements. Indeed, Newton's theory of motion has much in common with a conventional axiomatic system, such as Euclidean geometry. Like all axiomatic systems, Newtonian mechanics starts from a set of terms that are undefined within the system. In this case, the fundamental terms are mass, position, time, and force. It is taken for granted that we understand what these terms mean, and, furthermore, that they correspond to measurable quantities that can be ascribed to, or associated with, objects in the world around us. In particular, it is assumed that the ideas of position in space, distance in space, and position as a function of time in space, are correctly described by conventional Euclidean vector algebra and vector calculus. The next component of an axiomatic system is a set of axioms; this is a set of unproven propositions, involving the undefined terms, from which all other propositions in the system can be derived via logic and mathematical analysis. In the present case, the axioms are called Newton's laws of motion, and can only be justified via experimental observation. Note, incidentally, that Newton's laws, in their primitive form, are only applicable to point objects (i.e., objects of negligible spatial extent). However, these laws can be applied to extended objects by treating them as collections of point objects.

One difference between an axiomatic system and a physical theory is that, in the latter case, even if a given prediction has been shown to follow necessarily from the axioms of the theory, it is still incumbent upon us to test the prediction against experimental observations. Lack of agreement might indicate faulty experimental data, faulty application of the theory (for instance, in the case of Newtonian mechanics, there might be forces at work that we have not identified), or, as a last resort, incorrectness of the theory. Fortunately, Newtonian mechanics has been found to give predictions that are in excellent agreement with experimental observations in all situations in which it would be expected to hold.

In the following, it is assumed that we know how to set up a rigid Cartesian frame of reference, and also how to measure the positions of point objects as functions of time within this frame. In addition, it is taken for granted that we have some basic familiarity with the laws of mechanics.