Numerical Solutions of Classical Equations of Motion

Exact analytical solutions of the equations of motion exist only for simple systems, of the types that are discussed in elementary classical mechanics courses, and therefore numerical integration methods are very important in practice. Here we will discuss some commonly used differential equation solvers and use them to study the dynamics of mechanical systems, including ones that exhibit chaotic dynamics. We will limit the discussion to a single moving body, although the methods can be easily generalized to many-body systems as well—dynamics of many-body systems will be discussed later in connection with molecular dynamics simulations.