.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/classical/hamiltonian/plot_01_henon_heiles.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_classical_hamiltonian_plot_01_henon_heiles.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_classical_hamiltonian_plot_01_henon_heiles.py:


The Henon-Heiles system: Poincare sections and KAM torus breakdown
========================================================================

:class:`~physicskit.classical.systems.hamiltonian.HenonHeilesSystem`
is the classic non-integrable 2-DOF Hamiltonian (originally a model of
stellar orbits in an axisymmetric galactic potential),

.. math::

    H = \frac{p_x^2 + p_y^2}{2} + \frac{x^2 + y^2}{2}
        + x^2 y - \frac{y^3}{3} ,

evolved here at three fixed energies :math:`E = H`, each with several
trajectories launched from different starting angles on the
:math:`y = 0` line. Below :math:`E \approx 1/6` the :math:`(x, p_x)`
Poincare section at :math:`y = 0,\ p_y > 0` is dominated by smooth
invariant curves (KAM tori: nearly every trajectory is
quasi-periodic). As :math:`E` approaches and exceeds :math:`1/6`, more
and more of those tori break up into a chaotic sea -- one of the
classic numerical demonstrations of the KAM theorem's limits.

.. GENERATED FROM PYTHON SOURCE LINES 25-53



.. image-sg:: /api/gallery/classical/hamiltonian/images/sphx_glr_plot_01_henon_heiles_001.png
   :alt: Poincare sections (y=0, py>0): smooth tori give way to chaos as E rises, Poincare section, Poincare section, Poincare section
   :srcset: /api/gallery/classical/hamiltonian/images/sphx_glr_plot_01_henon_heiles_001.png
   :class: sphx-glr-single-img





.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.classical.systems.hamiltonian import HenonHeilesSystem
    from physicskit.classical.visualizers.phase_space import plot_poincare_section

    energies = [0.05, 0.1167, 0.16]
    fig, axes = plt.subplots(1, 3, figsize=(13, 4.3), sharex=True, sharey=True)

    rng = np.random.default_rng(0)
    for ax, E in zip(axes, energies):
        ax.set_title(f"E = {E:.4f}", fontsize=14)
        # several trajectories per energy, different starting angles on the y=0 line
        for x0 in np.linspace(-0.3, 0.3, 6):
            v2 = 2 * E - x0**2  # from H = 0.5*(px^2+py^2) + 0.5*x^2 at y=0
            if v2 <= 0:
                continue
            px0 = np.sqrt(v2) * rng.choice([-1, 1]) * 0.6
            py0 = np.sqrt(max(v2 - px0**2, 1e-6))
            system = HenonHeilesSystem(np.array([x0, 0.0]), np.array([px0, py0]))
            result = system.integrate((0, 400), dt=0.01, method="yoshida4")
            plot_poincare_section(result.q[:, 0], result.p[:, 0], result.q[:, 1], result.p[:, 1], value=0.0, direction=1, ax=ax, color="steelblue")
        ax.set_xlabel("x", fontsize=12)
    axes[0].set_ylabel("px", fontsize=12)
    fig.suptitle("Poincare sections (y=0, py>0): smooth tori give way to chaos as E rises", fontsize=16, fontweight="bold")
    fig.tight_layout(rect=[0, 0, 1, 0.93])

    plt.show()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 1.387 seconds)


.. _sphx_glr_download_api_gallery_classical_hamiltonian_plot_01_henon_heiles.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_henon_heiles.ipynb <plot_01_henon_heiles.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_henon_heiles.py <plot_01_henon_heiles.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_henon_heiles.zip <plot_01_henon_heiles.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_