The double pendulum is just like a pendulum, but doubled. Ha. No, it’s a pendulum on the end of a pendulum. You often see this in your classical mechanics class — but the solution isn’t exactly trivial. It is also used as an example of chaotic motion. Finally, one more comment before we get into the details. Although the double pendulum is fairly popular, it’s actually not so trivial to build a real life version. You can’t just get two masses and strings because it won’t stay in a 2D plane of motion.
OK, so here is a double pendulum.
I’ve jumped the gun and went ahead with two variables for the position of the pendulum. Here you can see I’m using θ1 and θ2. I guess I should explain WHY I’m doing it that way. Oh, also I’m calling the lengths of the two strings as R1 and R2 with masses m1 and m2 (remember, Medium doesn’t do subscripts — sorry about that but it’s not my fault). We can make things easier later by using R1 = R2 and m1 = m2, but I’m going to do this generically.
We Can’t Use Newtonian Mechanics
Let me start with a single mass on a string — a single pendulum (though everyone just…
