Double Pendulum Analysis

A Python simulation of the double pendulum system using Lagrangian mechanics to demonstrate chaotic behavior and analyze normal modes.

Overview

This repository contains a numerical investigation of the double pendulum. It explores two main regimes:

1. Non-Linear (Chaotic): Solving the full Euler–Lagrange equations to visualize sensitivity to initial conditions.
2. Linearized (Small Angle): Deriving the equations of motion for small angles to find normal-mode frequencies (eigenvalues) and verifying them using Fast Fourier Transforms (FFT).

Demo

Figure 1: Simulation of the chaotic regime showing the sensitivity to initial conditions.

Repository Structure

File	Description
notebooks/01_full_simulation.ipynb	Derives the full non-linear EOMs using SymPy, solves them with scipy.integrate.solve_ivp, and animates the chaotic trajectory.
notebooks/02_linearization_analysis.ipynb	Linearizes the system around the stable equilibrium, calculates eigenvalues/eigenfrequencies, and compares analytical predictions with FFT results.

The Physics

The system consists of two masses $m_1$ and $m_2$ connected by massless rods of length $l_1$ and $l_2$. The configuration is described by the angles $\theta_1$ and $\theta_2$.

Lagrangian Formulation

The Lagrangian $\mathcal{L} = T - V$ is:

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

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

Linearization & Normal Modes

For small angles ($\cos\theta \approx 1$, $\sin\theta \approx \theta$), the system behaves as two coupled harmonic oscillators.

The normal-mode frequencies follow from:

$$ \det(K - \omega^2 M) = 0 $$

Where:

• $M$ is the mass matrix
• $K$ is the stiffness matrix from the linearized potential energy

Installation

1. Clone the repository:

   git clone https://github.com/abdelrahman-26/double-pendulum-analysis.git
   

2. Install dependencies:

   pip install numpy scipy matplotlib sympy jupyter
   

Usage

Launch Jupyter Notebook inside the repository folder:

jupyter notebook

Then open the notebooks in the notebooks/ directory:

• Full simulation: 01_full_simulation.ipynb
  Run all cells to see the animation of the chaotic system.

• Physics + FFT analysis: 02_linearization_analysis.ipynb
  Walks through the small-angle approximation, eigenvalue calculation, and FFT analysis.

Key Libraries

• SymPy — symbolic derivation of Euler–Lagrange equations
• SciPy — solve_ivp for numerical integration.
• NumPy — array operations, FFT
• Matplotlib — plotting and animation

License

Distributed under the MIT License. See LICENSE for details.

Acknowledgments

Derived from standard Lagrangian mechanics coursework.
Visualizations inspired by examples from the Python scientific computing community.