A double pendulum is an example of a chaotic system. It consists of a simple pendulum attached to the end of another simple pendulum. This simple system demonstrates complicated, chaotic behaviour and provides a simple example of complex systems. One of the main characteristics of chaotic systems is the sensistivity to initial conditions. Small variations in the initial poisition and velocity will produce radically different behaviour.
The position and angle of each pendulum is described using the Lagrangian $$\mathcal{L} = \frac{1}{2}\left(m \left(\frac{dx_1}{dt}^2 + \frac{dx_2}{dt}^2 + \frac{dy_1}{dt}^2 + \frac{dy_2}{dt}^2\right) + I\left(\frac{d\theta_1}{dt}^2 + \frac{d\theta_2}{dt}^2 \right)\right) - mg(y_1+y_2)$$ where the first term describes the kinetic energy dependent on mass m and the respective velocities. The second term describes the rotational kinetic energy dependent on the moment of inertia I and the change in the angle of the pendulums. Finally, the potential energy dependent on the mass, gravity and the physical position of the pendulums. This gives the equations of motion $$\frac{dp_{\theta_1}}{dt} = \frac{\partial \mathcal{L}}{\partial \theta_1} = -\frac{1}{2}ml^2\left(\frac{d\theta_1}{dt}\frac{d\theta_2}{dt}\sin(\theta_1-\theta_2) +3\frac{g}{l}\sin\theta_1\right)$$ $$\frac{dp_{\theta_2}}{dt} = \frac{\partial \mathcal{L}}{\partial \theta_2} = -\frac{1}{2}ml^2\left(-\frac{d\theta_1}{dt}\frac{d\theta_2}{dt}\sin(\theta_1-\theta_2) +\frac{g}{l}\sin\theta_2\right)$$ where l is the total length of the pendulum. Using the simulation below, investigate the following problems.