Introduction

Abstract

The foundations of the part of mechanics that deals with the motion of point-particles were laid by Newton in 1687 in his Philosophiae Naturalis Principia Mathematica. This classic work does not consist of a carefully thought-out system of axioms in the modern sense, but rather of a number of statements of various significance and generality, depending on the state of knowledge at the time. We shall take his second law as our starting point: “Force equals mass times acceleration.” Letting x_i(t) be the Cartesian coordinates of the i-th particle as a function of time, this means

$${m_i}\frac{{{d^2}{x_i}(t)}}{{d{t^2}}}=\,{F_i}\left({x{}_i}\right),\,\,\,i =\,1,\,2,\,...,\,N,$$

((1.1.1))

where Fi denotes the force on the i-th particle. In nature, so far as we know, there are just four fundamental forces: the strong, weak, electromagnetic, and gravitational forces. In physics books there are in addition numerous other forces, such as friction, exchange forces, forces of constraint, fictitious forces (centrifugal, etc.), and harmonic forces, with which we shall only be peripherally concerned. The first two fundamental forces operate at the subatomic level, outside the realm of classical mechanics, so in fact we shall only discuss gravitation and electromagnetism.