Dynamics of particles

Abstract

The use of Newton’s laws to study the motion of a particle in a given field of force is essentially a mathematical rather than a physical question. If the field of force is defined as a function of position by the specification of F(r), the problem is to solve the equation mȑ = F(r). With modern computers, it is easy to trace out the trajectory of a particle with any desired precision, and if the need is simply to know the trajectory, given the velocity of the particle at the starting point, there is little point usually in doing anything except feed the information to a computer and await the answer. However, every initial velocity results in a different trajectory, and computing many trajectories may be an uneconomical way of finding out whether the trajectories are governed by any general rules. This is not to suggest that computers should not be used in some circumstances; indeed, a few experimental trajectories traced out with the help of a computer may be the way of guessing what general rules may exist. But ultimately a certain amount of mathematical analysis is the surest way of revealing the regularities and necessary properties of a family of trajectories. We shall study here the analysis of a particular type of trajectory, that executed by a particle in a central field of force F(r) = rf(r), since this has certain simple rules to obey, and moreover there are particular laws of force, such as the harmonic force, f(r) = a negative constant, and the inverse-square law, f(r) ɷ r-3 (note the presence of r in the definition of the force law; in these two cases F varies as r and as r-2 respectively) which occur in important contexts.