Double Pendulum

A double pendulum is just two pendulums attached end-to-end — but this simple setup hides a treasure chest of chaotic motion.

• The first pendulum swings freely from a pivot.

• The second pendulum hangs from the end of the first.

• When you start them swinging, the motion quickly becomes unpredictable — tiny differences in starting conditions can lead to vastly different results.
  This is a textbook example of deterministic chaos: the system follows the laws of physics exactly, but predicting it for long periods is nearly impossible without extreme precision.

2. The Physics & Math

We’ll assume:

• $m_1, m_2$ = masses of the first and second pendulum bobs

• $l_1, l_2$ = lengths of the first and second rods

• $\theta_1, \theta_2$ = angles from vertical

• $g$ = acceleration due to gravity

The dynamics are derived from Lagrangian mechanics, which considers the difference between kinetic and potential energy:

Kinetic Energy:

$$T = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2$$

with $v_1, v_2$ from rotational motion.

Potential Energy:

$$V = -m_1 g l_1 \cos\theta_1 - m_2 g \left( l_1 \cos\theta_1 + l_2 \cos\theta_2 \right)$$

The Lagrangian is:

$$L = T - V$$

Applying the Euler–Lagrange equations yields the coupled nonlinear ODEs:

$$\ddot{\theta}_1 = \frac{-g (2 m_1 + m_2) \sin\theta_1 - m_2 g \sin(\theta_1 - 2\theta_2) - 2 \sin(\theta_1 - \theta_2) m_2 \left( \dot{\theta}_2^2 l_2 + \dot{\theta}_1^2 l_1 \cos(\theta_1 - \theta_2) \right)} {l_1 \left[ 2 m_1 + m_2 - m_2 \cos\left( 2\theta_1 - 2\theta_2 \right) \right]}$$$$\ddot{\theta}_2 = \frac{ 2 \sin(\theta_1 - \theta_2) \left[ \dot{\theta}_1^2 l_1 (m_1 + m_2) + g (m_1 + m_2) \cos\theta_1 + \dot{\theta}_2^2 l_2 m_2 \cos(\theta_1 - \theta_2) \right]} {l_2 \left[ 2 m_1 + m_2 - m_2 \cos\left( 2\theta_1 - 2\theta_2 \right) \right]}$$

That’s the complete set of equations — they’re messy, but they describe every twist and turn.

3. Things to Try in the Simulation

Initial Conditions:

• Start both pendulums in the same direction and give them a gentle nudge.

• Start one pendulum vertical and the other horizontal — watch how energy transfers.

• Change the starting angles by just 0.001 radians — observe how quickly they diverge.

Parameters:

• Change $m_1$ or $m_2$ — heavier lower mass often makes motion wilder.

• Try different rod lengths $l_1, l_2$ — longer rods swing slower.

• Set gravity to lower values — you can simulate “Moon pendulums.”

Energy Checks:

• Track total energy $E = T + V$ to see if your simulation conserves energy.

4. Why It’s Cool

• Chaos in action: The double pendulum is a beautiful intro to chaotic dynamics.

• Physics + math synergy: Lagrangian mechanics is more elegant than Newton’s laws here.

• Practical applications: Understanding coupled oscillators shows up in robotics, structural engineering, and even molecular vibrations.