.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/special_functions/elliptic/plot_02_jacobi_elliptic_functions.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_special_functions_elliptic_plot_02_jacobi_elliptic_functions.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_special_functions_elliptic_plot_02_jacobi_elliptic_functions.py:


Jacobi elliptic functions and the pendulum
================================================

Plots sn, cn, dn for m = 0.9, showing their real period 4K(m), and uses
them to write the exact motion of a large-amplitude pendulum.

.. GENERATED FROM PYTHON SOURCE LINES 10-15

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.special_functions import complete_elliptic_integral_first_kind, jacobi_elliptic_functions








.. GENERATED FROM PYTHON SOURCE LINES 16-18

sn, cn, dn over two periods
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 18-34

.. code-block:: Python


    m = 0.9
    K = complete_elliptic_integral_first_kind(m)
    u = np.linspace(0, 8 * K, 800)
    r = jacobi_elliptic_functions(u, m)

    fig, ax = plt.subplots()
    ax.plot(u / K, r.sn, label="sn")
    ax.plot(u / K, r.cn, label="cn")
    ax.plot(u / K, r.dn, label="dn")
    ax.set_xlabel("u / K(m)")
    ax.set_title(f"Jacobi elliptic functions, m = {m}")
    ax.legend()
    print(f"K({m}) = {K:.6f}; sn(K) = {jacobi_elliptic_functions(K, m).sn:.12f}")
    print(f"max |sn^2 + cn^2 - 1| = {np.max(np.abs(r.sn**2 + r.cn**2 - 1)):.1e}")




.. image-sg:: /api/gallery/special_functions/elliptic/images/sphx_glr_plot_02_jacobi_elliptic_functions_001.png
   :alt: Jacobi elliptic functions, m = 0.9
   :srcset: /api/gallery/special_functions/elliptic/images/sphx_glr_plot_02_jacobi_elliptic_functions_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    K(0.9) = 2.578092; sn(K) = 1.000000000000
    max |sn^2 + cn^2 - 1| = 2.2e-16




.. GENERATED FROM PYTHON SOURCE LINES 35-37

Exact pendulum motion: sin(theta/2) = k sn(K - t, k^2) with omega_0 = 1
------------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 37-51

.. code-block:: Python


    theta0 = np.radians(150.0)
    k = np.sin(theta0 / 2)
    K_pend = complete_elliptic_integral_first_kind(k**2)
    t = np.linspace(0, 4 * K_pend, 400)
    theta = 2 * np.arcsin(k * jacobi_elliptic_functions(K_pend - t, k**2).sn)
    print(f"\n150 degree pendulum: period 4K = {4 * K_pend:.4f}, small-angle period 2 pi = {2 * np.pi:.4f}")

    fig, ax = plt.subplots()
    ax.plot(t, np.degrees(theta), label="exact (Jacobi sn)")
    ax.plot(t, np.degrees(theta0 * np.cos(t)), "--", label="small-angle approximation")
    ax.set_xlabel(r"$\omega_0 t$")
    ax.set_ylabel("angle (degrees)")
    ax.legend()



.. image-sg:: /api/gallery/special_functions/elliptic/images/sphx_glr_plot_02_jacobi_elliptic_functions_002.png
   :alt: plot 02 jacobi elliptic functions
   :srcset: /api/gallery/special_functions/elliptic/images/sphx_glr_plot_02_jacobi_elliptic_functions_002.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    150 degree pendulum: period 4K = 11.0723, small-angle period 2 pi = 6.2832

    <matplotlib.legend.Legend object at 0x11cf434d0>




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

   **Total running time of the script:** (0 minutes 0.055 seconds)


.. _sphx_glr_download_api_gallery_special_functions_elliptic_plot_02_jacobi_elliptic_functions.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_jacobi_elliptic_functions.ipynb <plot_02_jacobi_elliptic_functions.ipynb>`

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

      :download:`Download Python source code: plot_02_jacobi_elliptic_functions.py <plot_02_jacobi_elliptic_functions.py>`

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

      :download:`Download zipped: plot_02_jacobi_elliptic_functions.zip <plot_02_jacobi_elliptic_functions.zip>`


.. only:: html

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

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