Elastic Pendulum Chain Simulation

This project was carried out as part of the Theoretical Physics course. The technologies and numerical methods used in the project were initiated by Dr. Andrzej Kądzielawa.

This project simulates a chain of N elastic particles (spring-mass system) swinging under gravity. It uses symbolic mathematics to derive the exact equations of motion (Lagrangian formalism), generates optimized C code for numerical solving, and visualizes the system in real-time.

Features

• N-Particle Chain: Simulate any number of connected particles forming a chain.
• 2D and 3D Modes: Visualize the chain swinging in a 2D plane or in full 3D space.
• High Performance:
  • SymPy derives the Lagrangian equations automatically.
  • C Code Generation: The physics equations are converted to C code on-the-fly.
  • RK4 Solver: A 4th-order Runge-Kutta integrator written in C handles the time-stepping.
• Visualization: Smooth animation using matplotlib.

Requirements

• OS: Linux/Unix (required for GCC and .so shared object generation).
• Compiler: GCC.
• Python: 3.x
• Dependencies: numpy, matplotlib, sympy.

Setup

1. Create a virtual environment (recommended):

   python3 -m venv venv_matplotlib
   source venv_matplotlib/bin/activate

2. Install dependencies:

   pip install numpy matplotlib sympy

How to Run

Navigate to the project root directory. The main script is run/particles.py.

1. Basic 2D Chain

To simulate a single elastic pendulum (or default number) in 2D:

python run/particles.py

2. Simulation with N Particles

To simulate a chain of N particles (e.g., 4 particles), pass the number as an argument:

python run/particles.py 4

3. 3D Simulation

To run the simulation in 3D mode, add the 3d argument. You can combine this with the number of particles:

# 3D simulation with 3 particles
python run/particles.py 3 3d

(Note: The particles are initialized with slight offsets to induce motion in all three dimensions.)

Project Structure

• run/: Python source files.
  • particles.py: Main entry point. Orchestrates physics generation, compilation, and animation.
  • animation.py: Handles 2D and 3D plotting/animation.
  • lagrangian.py: Generates C code from SymPy Lagrangian expressions.
  • ccompiler.py: Compiles C source files into shared libraries.
  • cprototype.py: CTypes wrappers for communicating with the C libraries.
• solver/: C source files and compiled libraries.
  • solver.c: Implementation of the Runge-Kutta 4 (RK4) solver.
  • generated_lagrangian.c: Auto-generated C code containing the specific equations of motion for your chosen N.
  • libsolver.so, liblagrangian.so: Compiled shared libraries.

Troubleshooting

• Compilation Errors: Ensure you have gcc installed and accessible in your system PATH.
• Visualization Issues: If running in a headless environment (like a server or some containers), the window will not appear. Ensure you have an X server running or use X11 forwarding if accessing remotely.