This page will briefly describe how to use this tool to propagate the accelerations for a simple double pendulum. This tutorial considers the motion of a double pendulum constrained to a plane. To follow the motion of the double pendulum, we must understand the time dependence of the two angles θ1 and θ2. This means there are two dimensions to the problem. θ1 will be our x coordinate and θ2 will be our y coordinate. If you have not already done so, launch the Newton Tool.
Acceleration Equations
Enter 2 in the number of dimensions field in the Acceleration(s) panel. Two fields for the accelerations will appear. Notice the variables used are x, y, vx, vy, ax, and ay.
The acceleration equations for this example are expressed in θ1 and θ2.
These equations are complex, so for simplicity assume g=10, L2=1, L1=1, m1=m2=1. With these assumptions, the accelerations reduce to the ones below. Carefully cut and paste these into the fields in the Acceleration Panel then click next.
x acceleration
-(20*SIN(x)+SIN(x-y)*(vy^2+vx^2*COS(x-y))-10*SIN(y)*COS(x-y))/(2-(COS(x-y))^2)
y acceleration
(2*vx^2*SIN(x-y)+COS(x-y)*(vy^2*SIN(x-y)+20*SIN(x))-20*SIN(y))/(2-(COS(x-y))^2)
It's very important to notice that x and y are no longer the Cartesian coordinates x and y, but rather are two variables that represent the angle a pendulum makes with a vertical axis. These coordinates are consequently measured in radians.
Initial Conditions
On the initial conditions panel set the initial x to be 0.6 and the initial y to be 2.0. These values are in radians.
Plots
Below are a couple of examples of some graphs easily generated from the above equations.
y = θ2 vs. time
y = θ2 vs. x = θ1
θ2 acceleration vs. θ1 acceleration
Movies
One may use the data generated from the Newton Tool and IDL (or other software) to generate movies of the motion of the pendula themselves or of the motion in θ1 vs θ2.
Challenge
The double pendulum is a rich dynamical system. For example, here is a movie generated from data calculated by the Newton Tool. It shows the pendula for a case in which the initial conditions cause "flipping"! The double pendulum can exhibit classical chaos. Can you vary the initial conditions to study the chaotic nature of the system?
