Double Pendulum Behavior Analysis

Overview

This project investigates the complex dynamics of a double pendulum system. It explores how simple mechanical systems can exhibit chaotic behavior and extreme sensitivity to initial conditions.

Objectives

• Derive equations of motion using Lagrangian mechanics
• Simulate the system's trajectory over time
• Analyze the onset of chaos and Lyapunov exponents
• Visualize phase space trajectories

Approach

Equations of motion were derived using the Euler-Lagrange equation. The resulting system of differential equations was solved numerically using Runge-Kutta methods. The simulation tracked the position and velocity of both pendulum bobs to generate trajectory plots and Poincaré maps.

Results

The analysis confirmed that while the system is deterministic, it becomes unpredictable over long timescales due to chaos. The project produced compelling visualizations of fractal-like patterns in phase space and demonstrated the butterfly effect in mechanical systems.