Chaotic Pendulum Simulation

An interactive chaotic pendulum simulation showcasing the double pendulum system and chaotic dynamics.

The double pendulum, or chaotic pendulum, is a system that comprises of a pendulum attached to the end of another pendulum. This system is governed by a set of coupled non-linear ordinary differential equations that cannot be solved analytically. In my simulation I implement fourth-order Runge-Kutta (RK4) integration to numerically solve the differential equations. The website showcases the derivation of the equations of motion using the Lagrangian formalism as well as phase-portraits for the system.

Theory

The generalized coordinates of the system are the angles of the two rods relative to their vertical.

The kinetic and potential energy of the system is:

$$T = \frac{1}{2}ML_1^{2}\dot{\theta_1}^{2} + \frac{1}{2}m_2L_2^{2}\dot{\theta_2}^{2} + m_2L_1L_2\dot{\theta_1}\dot{\theta_2}\cos\Delta\theta$$

$$U = -MgL_1\cos\theta_1 - m_2gL_2\cos\theta_2$$

The Lagrangian of the system is:

$$\mathcal{L} = \frac{1}{2}ML_1^{2}\dot{\theta_1}^{2} + \frac{1}{2}m_2L_2^{2}\dot{\theta_2}^{2} + m_2L_1L_2\dot{\theta_1}\dot{\theta_2}\cos\Delta\theta + MgL_1\cos\theta_1 + m_2gL_2\cos\theta_2$$

We then apply the Euler-Lagrange equation to our generalized coordinates:

$$\frac{\partial L}{\partial q} - \frac{d}{dt} \left(\frac{\partial L}{\partial \dot{q}}\right) = 0$$

After applying the Euler-Lagrange equation we do some simplifying (check out the website) to arrive at our equations of motion. The equations of motion are a set of coupled non-linear ODEs and have NO analytical solution 😿. Thus, they must be solved via numerical methods. In my simulation I use fourth-order Runge-Kutta (RK4) integration.