Orbits of Particles

Abstract

If a system has only a very few degrees of freedom its equations of motion are often obvious. One writes down Newton’s laws and sets about solving them. If they are not easy to solve it is because some equations are difficult. In complicated systems, even formulating the equations presents some problems, and most of this book is devoted to more advanced and general ways of writing down such equations and constructing their exact and approximate solutions. This chapter exhibits some relatively simple systems chosen so as to illustrate various forms of dynamical behavior. The first few examples need only Newton’s laws and a little mathematics, but for the later ones it will be convenient to introduce new variables and write equations of motion in terms of generalized coordinates. This will be done using Lagrange’s technique, which leads not only to general and useful equations of motion but also to the powerful methods of solution that follow.