For the last 400 years (since Galileo) the pendulum has been a source of intrigue and insight for physicists. We’ll look at the pendulum in this article and the next, in ever more complicated variations. This becomes fairly mathematical at points, the thing is, when you *can* read them the equations are very illuminating and when you can’t they tend to make the rest of the text unreadable. If you can’t follow the equations skip them and appreciate them as pieces of abstract art that break up the paragraphs, the text is enough. If you don’t even read the text here are the instructions: click the mouse in the box and drag the red bob to its start position, click again and drag the blue bob then release.

Let’s start with the basics: Newton’s second law is a differential equation,
In this form we can make a very simple numerical solution, the idea is that if you zoom in far enough any curve looks like a line. The quantity $\frac{d\vec{y}(t)}{dt}$ means the slope of the curve $\vec{y}(t)$ at $t$. So we approximate the curve by a line and glue a bunch of these lines together. Mathematically, let $h$ be a very small number then Newton’s equation becomes, |

Doing a little vector analysis, the pendulum moves on a circle of length $l$. The pendulum bob has traveled a distance $l \theta$ along the circle when the force on it is $-mg \sin\theta$, tangent to the circle. Newton’s equations give, \[ \begin{array}{c} m l \frac{d^2 \theta}{dt^2} = -m g \sin\theta\\ \frac{d^2 \theta}{dt^2} + \frac{g}{l} \sin\theta = 0 \end{array} \] If you’ve solved this in a physics class you made an approximation here that the pendulum moves horizontally, otherwise the integrals are too difficult. But, because we’re solving this numerically, we don’t have to make that approximation so our pendulum does indeed move on a circle. Click the animation below and drag the bob to somewhere in the box and let go. The red pendulum is the result of the method above, known as the Euler method.
The Euler method is a bad method, errors accumulate quite quickly, its advantage is how easy it is to program. The standard compromise between improvement and complexity is the 4${}^{th}$ order Runge-Kutta method. Runge from the Runge-Lenz vector and Kutta from the Kutta-Joukowski airfoil. This method is basically equivalent to Simpson’s Rule. Explicitly, Given a decent algorithm for solving differential equations we use it for something more interesting above, the double pendulum. The double pendulum is one of the quintessential chaotic systems, displaying extreme sensitivity to initial conditions. You can start the pendulum watch it for a while, then try to restart it in the same position. Unless you click on the exact same pixels the motion will be completely different after a few moments of swinging. Take that as a challenge if you like! |