Three-Body Problem Simulation in Python

This repository demonstrates how to simulate and analyze the three-body problem using Python. The three-body problem involves predicting the motion of three massive bodies under their mutual gravitational attraction. It is a fascinating problem in physics and mathematics, often exhibiting chaotic behavior and sensitivity to initial conditions.

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Features

• Mathematical Overview: Detailed explanation of the equations of motion governing the three-body problem.
• Numerical Integration: Implementation using Python's scipy.integrate.solve_ivp to solve the coupled ordinary differential equations (ODEs).
• Visualization: Interactive plots of the trajectories of the three bodies.
• Example Simulation: Includes a simplified simulation of the Sun-Earth-Moon system.

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Requirements

Ensure you have Python 3.8+ installed along with the following packages:

• numpy
• scipy
• matplotlib

To install the dependencies, use:

pip install numpy scipy matplotlib

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Repository Contents

• three_body_problem.py: The main Python script containing the implementation of the simulation.
• README.md: This file, providing an overview of the project.
• example_plot.png: A sample visualization of the three-body trajectories.

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Usage

1. Clone this repository:

git clone https://github.com/jman4162/three-body-problem.git
cd three-body-problem

2. Run the simulation script:

python three_body_problem.py

3. View the plotted trajectories of the three bodies in a 2D plane.

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Highlights of the Implementation

Mathematical Background

The motion of the three bodies is governed by Newton's laws of gravitation and motion. For bodies with masses $m_1$, $m_2$, and $m_3$, the equations of motion are expressed as:

$$ \mathbf{a}_1 = -G m_2 \frac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3} - G m_3 \frac{\mathbf{r}_1 - \mathbf{r}_3}{|\mathbf{r}_1 - \mathbf{r}_3|^3}. $$

(Refer to the three_body_problem.py script for more details.)

Numerical Integration

The simulation uses Python's solve_ivp function to integrate the ODEs over time. Adaptive time-stepping methods ensure accuracy and efficiency.

Visualization

The script generates a plot of the trajectories of the three bodies in a 2D plane using Matplotlib. Example:

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Example: Sun-Earth-Moon System

The repository includes an example simulation where:

• The Sun, Earth, and Moon are modeled as point masses.
• Initial conditions are chosen to approximate their real-world positions and velocities.

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Customization

You can customize the following parameters in three_body_problem.py:

• Masses: Modify m_sun, m_earth, m_moon.
• Initial Conditions: Change the starting positions and velocities of the bodies.
• Time Span: Adjust the duration of the simulation (t_span).

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Learning Objectives

By exploring this repository, you will:

• Gain insights into the physics of the three-body problem.
• Learn how to use Python to numerically solve ODEs.
• Understand the sensitivity and chaotic nature of multi-body gravitational systems.

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Contributing

Contributions are welcome! If you find bugs, want to add features, or improve the code, please open an issue or submit a pull request.

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License

This project is licensed under the MIT License. See LICENSE for details.

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Acknowledgments

This project is inspired by the fascinating problem of multi-body dynamics in classical mechanics. The numerical methods and visualization techniques used are based on modern computational tools in Python.