TABLE OF CONTENTS In this chapter we apply phase plane ideas to various one-dimensional systems that model a particle moving under Newton s laws of motion. First we consider systems in which the energy is constant, and then we consider systems in which there is some dissipation.
We consider a particle of mass m moving on a line in a potential force field, such that its potential energy at position x is given by V ( x). Then its kinetic energy is 1/2 m 
2, and its total energy is
If the energy is conserved then we can differentiate to give
provided that ? 0 we can cancel this term and obtain [1]
By setting y = , we can rewrite this as the coupled system
In all that follows we will take m = 1 for simplicity.
If you think of these equations as describing the motion of a bead sliding on a wire whose height at coordinate x is given by V ( x) you will get the correct qualitative idea of how the solutions should behave, although this interpretation is not entirely accurate, as discussed in the next section.
Because of the relatively simple form of these equations the possible behaviour of the solutions is restricted. First, note that at any stationary point ( x* , y*) we must have y* = 0 (zero velocity), and x* must be a turning point...
