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Example 02: Double Pendulum - Chaos Theory in Action

Where deterministic physics meets the unpredictable!

Butterfly Effect

📖 Overview

The double pendulum is a canonical example of deterministic chaos. The equations are completely known, yet long-term prediction is impossible because infinitely small differences in initial conditions lead to completely different outcomes.

"Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?"
— Edward Lorenz, 1972

Source: src/examples/02_double_pendulum/

🎯 Scenarios

Scenario 1: Chaotic Motion

Watch a single double pendulum evolve from a dramatic initial position (both arms at 135°).

Initial Conditions:
  θ₁ = 135° (upper arm)
  θ₂ = 135° (lower arm)
  ω₁ = ω₂ = 0 (released from rest)

Energy Conservation Check:
  Initial energy: -9.633814 J
  Final energy:   -9.633814 J
  Energy drift:   0.000001%   ← RK4 is EXCELLENT!

Outputs:

  • double_pendulum_angles.txt - θ₁(t) and θ₂(t)
  • double_pendulum_phase_space.txt - Chaotic attractor visualization
  • double_pendulum_trajectory.txt - Full trajectory for animation

Scenario 2: Butterfly Effect 🦋

THE ESSENCE OF CHAOS: Two pendulums starting with just 0.001° difference!

Pendulum A: θ₁ = 90.000°, θ₂ = 90°
Pendulum B: θ₁ = 90.001°, θ₂ = 90°
                    └── JUST 0.001° !!

Divergence Over Time:
  Time (s)    Angle Difference
  --------------------------------
      0.0           0.0010 deg
      2.0           0.0089 deg
      5.0           0.4721 deg
      8.0          47.2314 deg   ← 47,000× amplification!!!

The difference grows EXPONENTIALLY! This is the butterfly effect in action.

Scenario 3: Energy Conservation Check

Verify that numerical integration preserves the conserved quantities:

  • Total mechanical energy E = T + V should remain constant
  • RK4 typically achieves <0.001% drift over 10 seconds
  • This validates our numerical approach

⚛️ Physics

The System

A double pendulum consists of:

  • Mass 1 (m₁) at the end of rod of length l₁
  • Mass 2 (m₂) at the end of rod of length l₂, attached to m₁

Equations of Motion

From Lagrangian mechanics (L = T - V):

Angular accelerations:

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

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

These are nonlinear coupled ODEs — no closed-form solution exists!

Total Energy (Conserved)

$$E = \frac{1}{2}m_1 l_1^2 \dot{\theta}_1^2 + \frac{1}{2}m_2[l_1^2\dot{\theta}_1^2 + l_2^2\dot{\theta}_2^2 + 2l_1 l_2\dot{\theta}_1\dot{\theta}_2\cos(\theta_1-\theta_2)]$$ $$- (m_1+m_2)gl_1\cos\theta_1 - m_2 g l_2\cos\theta_2$$

🔧 MML Features Used

Feature Usage
IODESystem Define double pendulum dynamics
ODESystemFixedStepSolver Fixed-step integration
StepCalculators::RK4_Basic 4th-order Runge-Kutta
PolynomInterpRealFunc Smooth interpolation for plotting
SplineInterpParametricCurve Phase space trajectory
Visualizer Multi-function plotting

📁 Key Files

src/examples/02_double_pendulum/
├── main.cpp              # Demo scenarios
├── DoublePendulum.h      # Self-contained physics
└── CMakeLists.txt

🏃 Running

# Build
cmake --build build --target Example02_DoublePendulum

# Run
./build/src/examples/Release/Example02_DoublePendulum

📊 Sample Output

======================================================================
         MML DOUBLE PENDULUM - CHAOS DEMONSTRATION
======================================================================

   The double pendulum is a classic example of deterministic chaos.
   The motion is completely determined by the equations, yet
   impossible to predict long-term due to extreme sensitivity
   to initial conditions.

   "Does the flap of a butterfly's wings in Brazil set off
    a tornado in Texas?" - Edward Lorenz, 1972

======================================================================
  SCENARIO 1: Chaotic Double Pendulum Motion
======================================================================

Physical Parameters:
  Mass 1: 1 kg, Length 1: 1 m
  Mass 2: 1 kg, Length 2: 1 m

Initial Conditions:
  Theta 1: 135 degrees
  Theta 2: 135 degrees
  (Both arms pointing up-left - dramatic start!)

Solving for 10 seconds... Done! (2001 time steps)

Energy Conservation Check:
  Initial energy: -9.633814 J
  Final energy:   -9.633814 J
  Energy drift:   0.000001%

✓ Single trajectory complete!

🌀 What is Chaos?

Chaos ≠ Randomness!

Property Random Chaotic
Deterministic No Yes
Predictable (short-term) No Yes
Predictable (long-term) No No
Sensitive to initial conditions N/A EXTREMELY

Chaos is deterministic unpredictability. The equations completely determine the future, but we can never know initial conditions with infinite precision.

🎓 Learning Points

  1. Chaos from simplicity: Just 2 coupled oscillators → infinite complexity
  2. Exponential divergence: Errors grow as e^(λt) where λ is the Lyapunov exponent
  3. Phase space: The "attractor" shows structure in apparent randomness
  4. Energy conservation: Validates numerical accuracy despite chaotic behavior
  5. MML workflow: Same ODE framework handles simple and chaotic systems

🔗 Related Examples

  • Example 00: N-Body Gravity (another chaotic system!)
  • Example 04: Lorentz Transformations (Edward Lorenz discovered chaos!)

📚 References

  • Strogatz, Nonlinear Dynamics and Chaos - THE textbook on chaos
  • Lorenz (1963), "Deterministic Nonperiodic Flow" - Original chaos paper
  • Wikipedia: "Double Pendulum" - Good mathematical treatment