Order to Chaos in 2-Dimensional Motion

Reference and figures: Computational Physics, Koonin and Meredith, 1990.

A fundamental advantage of using computer in physics is the ability to study systems that cannot be solved analytically. In this section we are going to solve for the two-dimensional motion of a particle moving in a potential $V(x,y)$. We will assume that $V(x,y)$ is such as to confine the particle in a finite region of space at low energy. We will soon encounter such a system for which only numerical solutions exist.

The model Ordinary Differential Equations are derived from Newton equations

$$\frac{d x}{d t} = v_x$$

$$\frac{d y}{d t} = v_y$$

$$\frac{d v_x}{d t} = -\frac{1}{m} \frac{\partial V}{\partial x}$$

$$\frac{d v_y}{d t} = -\frac{1}{m} \frac{\partial V}{\partial y}$$

The solutions are specified by arbitrary initial conditions.

The fact that the motion of the particle derives from a potential implies that the total energy is a constant of the motion. Therefore the trajectories are restricted to a three-dimensional manifold. This might be the only statement applicable to all such systems.