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‎.gitignore

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# Byte-compiled / optimized / DLL files
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__pycache__/
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*.py[cod]
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*$py.class
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# C extensions
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*.so
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# Distribution / packaging
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.Python
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build/
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develop-eggs/
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dist/
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downloads/
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eggs/
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.eggs/
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lib/
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lib64/
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parts/
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sdist/
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var/
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wheels/
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share/python-wheels/
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*.egg-info/
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.installed.cfg
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*.egg
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MANIFEST
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# PyInstaller
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# Usually these files are written by a python script from a template
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# before PyInstaller builds the exe, so as to inject date/other infos into it.
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*.manifest
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*.spec
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# Installer logs
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pip-log.txt
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pip-delete-this-directory.txt
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# Unit test / coverage reports
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htmlcov/
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.tox/
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.nox/
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.coverage
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.coverage.*
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.cache
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nosetests.xml
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coverage.xml
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*.cover
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*.py,cover
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.hypothesis/
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.pytest_cache/
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cover/
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# Translations
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*.mo
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*.pot
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# Django stuff:
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*.log
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local_settings.py
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db.sqlite3
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db.sqlite3-journal
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# Flask stuff:
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instance/
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.webassets-cache
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# Scrapy stuff:
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.scrapy
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# Sphinx documentation
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docs/_build/
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# PyBuilder
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.pybuilder/
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target/
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# Jupyter Notebook
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.ipynb_checkpoints
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# IPython
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profile_default/
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ipython_config.py
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# pyenv
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# For a library or package, you might want to ignore these files since the code is
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# intended to run in multiple environments; otherwise, check them in:
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# .python-version
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# pipenv
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# According to pypa/pipenv#598, it is recommended to include Pipfile.lock in version control.
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# However, in case of collaboration, if having platform-specific dependencies or dependencies
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# having no cross-platform support, pipenv may install dependencies that don't work, or not
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# install all needed dependencies.
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#Pipfile.lock
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# UV
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# Similar to Pipfile.lock, it is generally recommended to include uv.lock in version control.
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# This is especially recommended for binary packages to ensure reproducibility, and is more
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# commonly ignored for libraries.
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#uv.lock
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# poetry
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# Similar to Pipfile.lock, it is generally recommended to include poetry.lock in version control.
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# This is especially recommended for binary packages to ensure reproducibility, and is more
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# commonly ignored for libraries.
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# https://python-poetry.org/docs/basic-usage/#commit-your-poetrylock-file-to-version-control
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#poetry.lock
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# pdm
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# Similar to Pipfile.lock, it is generally recommended to include pdm.lock in version control.
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#pdm.lock
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# pdm stores project-wide configurations in .pdm.toml, but it is recommended to not include it
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# in version control.
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# https://pdm.fming.dev/latest/usage/project/#working-with-version-control
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.pdm.toml
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.pdm-python
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.pdm-build/
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# PEP 582; used by e.g. github.com/David-OConnor/pyflow and github.com/pdm-project/pdm
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__pypackages__/
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# Celery stuff
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celerybeat-schedule
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celerybeat.pid
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# SageMath parsed files
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*.sage.py
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# Environments
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.env
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.venv
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env/
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venv/
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ENV/
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env.bak/
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venv.bak/
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# Spyder project settings
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.spyderproject
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.spyproject
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# Rope project settings
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.ropeproject
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# mkdocs documentation
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/site
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# mypy
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.mypy_cache/
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.dmypy.json
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dmypy.json
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# Pyre type checker
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.pyre/
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# pytype static type analyzer
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.pytype/
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# Cython debug symbols
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cython_debug/
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# PyCharm
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# JetBrains specific template is maintained in a separate JetBrains.gitignore that can
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# be found at https://github.com/github/gitignore/blob/main/Global/JetBrains.gitignore
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# and can be added to the global gitignore or merged into this file. For a more nuclear
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# option (not recommended) you can uncomment the following to ignore the entire idea folder.
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#.idea/
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# Ruff stuff:
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.ruff_cache/
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# PyPI configuration file
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.pypirc

‎README.md

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# Unified Framework for the Three-Body Problem
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[![Video](figures/animations/video_preview.png)](figures/animations/three_body_scenarios_comparative.mp4)
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This repository implements the unified theoretical framework that establishes rigorous isomorphisms between three distinct mathematical approaches to the three-body problem: Differential Galois Theory, Painlevé Analysis, and Quaternionic Regularization.
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## Overview
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The three-body problem—describing the motion of three bodies under mutual gravitational attraction—remains one of the fundamental challenges in mathematical physics. This implementation connects three mathematical approaches through precise isomorphisms:
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1. **Differential Galois Theory (DGT)**: An algebraic approach examining the structure of differential field extensions generated by solutions to variational equations.
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2. **Painlevé Analysis (PA)**: A complex-analytic approach examining the singularity structure of solutions.
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3. **Quaternionic Regularization (QR)**: A geometric approach that extends the domain of solutions from complex to quaternionic space.
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The key insight of this work is that these three perspectives are connected through precise mathematical isomorphisms, providing a powerful unified framework for analyzing the three-body problem.
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## Installation
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### Prerequisites
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The implementation requires Python 3.10+ and the following dependencies:
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```bash
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pip install numpy scipy sympy matplotlib pandas
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```
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### Clone the Repository
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```bash
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git clone https://github.com/username/three-body-isomorphisms.git
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cd three-body-isomorphisms
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```
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## Project Structure
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The implementation is divided into several modules:
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- `quaternion.py`: Implementation of quaternion algebra
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- `three_body_problem.py`: Implementation of the three-body problem, homothetic orbits, and Lagrangian solutions
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- `differential_galois.py`: Implementation of Differential Galois Theory analysis
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- `painleve_analysis.py`: Implementation of Painlevé Analysis
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- `quaternionic_regularization.py`: Implementation of Quaternionic Regularization methods
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- `isomorphism_verification.py`: Implementation of isomorphism verification
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- `kam_theory.py`: Implementation of KAM Theory integration
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- `visualization.py`: Implementation of visualization tools
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- `benchmark.py`: Implementation of benchmarking infrastructure
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## Running Tests
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Each module includes comprehensive tests that can be run individually to verify its functionality:
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```bash
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# Run tests for the quaternion module
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python quaternion.py
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# Run tests for the three-body problem module
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python three_body_problem.py
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# Run tests for the differential Galois theory module
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python differential_galois.py
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# Run tests for the Painlevé analysis module
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python painleve_analysis.py
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# Run tests for the quaternionic regularization module
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python quaternionic_regularization.py
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# Run tests for the isomorphism verification module
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python isomorphism_verification.py
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# Run tests for the KAM theory integration module
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python kam_theory.py
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# Run tests for the visualization module
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python visualization.py
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```
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You can also run a quick benchmark test to ensure the entire framework is functioning correctly:
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```bash
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python benchmark.py --test
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```
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## Running Benchmarks
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### Full Benchmark
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To generate all results, tables, and visualizations described in the paper:
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```bash
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python benchmark.py --output-dir results
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```
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This command will:
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1. Verify isomorphisms for homothetic orbits
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2. Verify isomorphisms for Lagrangian solutions
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3. Run benchmarks for both types of orbits
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4. Analyze isomorphism verification performance
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5. Generate KAM theory integration results
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6. Create visualizations and tables
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The results will be saved in the specified output directory (default: `results/`).
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### Verification Only
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To run only the verification benchmarks (faster):
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```bash
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python benchmark.py --verify-only --output-dir results
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```
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## Examining Results
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The benchmark process generates several types of outputs:
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### CSV Files
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- `homothetic_isomorphisms.csv`: Isomorphism verification results for homothetic orbits
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- `lagrangian_isomorphisms.csv`: Isomorphism verification results for Lagrangian solutions
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- `homothetic_performance.csv`: Performance benchmarks for homothetic orbits
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- `lagrangian_performance.csv`: Performance benchmarks for Lagrangian solutions
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- `verification_performance.csv`: Performance metrics for isomorphism verification
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- `isomorphism_kam_correspondence.csv`: Correspondence between isomorphisms and KAM theory
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- `kam_measure.csv`: KAM measure values for different mass parameters
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### LaTeX Tables
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- `table_homothetic_isomorphisms.tex`: Table of isomorphic structures in homothetic orbits
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- `table_lagrangian_isomorphisms.tex`: Table of quaternionic regularization and isomorphic structures for Lagrangian solutions
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- `table_verification_performance.tex`: Performance of isomorphism verification for near-exceptional case
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- `table_isomorphism_kam_correspondence.tex`: Correspondence between isomorphism structures and KAM theory
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### Visualizations
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- `homothetic_isomorphisms.png`: Parameter space of homothetic orbit isomorphisms
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- `lagrangian_isomorphisms.png`: Parameter space of Lagrangian solution isomorphisms
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- `homothetic_trajectory_*.png`: Trajectories for homothetic orbits
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- `lagrangian_trajectory_*.png`: Trajectories for Lagrangian solutions
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- `integration_diagram_*.png`: Visual representations of isomorphisms
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- `branching_structure_*.png`: Branching structures in the complex plane
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- `quaternionic_manifold_*.png`: Quaternionic branch manifolds
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- `kam_measure.png`: KAM measure vs. mass parameter
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## Key Mass Parameters
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The implementation pays special attention to three exceptional mass ratios that yield partially integrable systems:
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1. σ = 1/3
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2. σ = 2³/3³
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3. σ = 2/3²
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These exceptional cases exhibit specific isomorphism structures that are verified and analyzed throughout the implementation.
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## Interpreting the Results
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The implementation verifies the following key claims:
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1. **Three-Way Isomorphism**: The differential Galois group structure, Painlevé branching behavior, and quaternionic monodromy are isomorphic mathematical structures.
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2. **Unified Integrability Criterion**: The abelian nature of the differential Galois group, the Painlevé property, and trivial quaternionic monodromy are equivalent conditions for integrability.
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3. **Exceptional Mass Ratios**: The exceptional mass ratios (σ = 1/3, σ = 2³/3³, σ = 2/3²) exhibit specific isomorphism structures that correspond to partial integrability.
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4. **KAM Theory Integration**: The isomorphism structures are reflected in the measure of phase space occupied by KAM tori, with peaks at the exceptional mass ratios.

‎assets/A_Combined_Differential_Galois_Theory_and_Painlevé_Analysis_Approach_to_the_Three_Body_Problem.pdf

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