.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/classical/newtonian/plot_05_noether_hidden_symmetry.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_newtonian_plot_05_noether_hidden_symmetry.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_classical_newtonian_plot_05_noether_hidden_symmetry.py:


Noether's Theorem and the Kepler Problem's Hidden Symmetry
================================================================

Emmy Noether's theorem ties each continuous symmetry of a system's action
to an exactly conserved quantity built from it: time-translation symmetry
gives energy, rotational symmetry gives angular momentum. The Kepler
problem carries a third, less obvious conserved quantity, the
Laplace-Runge-Lenz vector

.. math::

    \mathbf{A} = \mathbf{p} \times L - \mu k \hat{\mathbf{r}}, \qquad
    L = x p_y - y p_x ,

fixed in both magnitude and direction only for the *exact* inverse-square
potential -- the classical fingerprint of a "hidden," purely dynamical
:math:`SO(4)` symmetry, with no equally obvious geometric transformation of
space and time behind it the way a rotation or translation is. Any purely
radial perturbation to :math:`V(r) = -k/r` leaves the *manifest*
rotational symmetry -- and hence angular momentum -- completely untouched
(a central force exerts no torque about its center regardless of its
radial profile), while still breaking the hidden symmetry and setting the
Laplace-Runge-Lenz vector adrift. Comparing all three conserved
quantities' drift, pure vs. perturbed, makes the distinction concrete:
some conservation laws come from symmetries obvious enough to write down
by inspection, others do not.

.. GENERATED FROM PYTHON SOURCE LINES 31-37

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.classical.systems.newtonian import KeplerSystem
    from physicskit.classical.utils.conservation import angular_momentum_drift, lrl_drift, relative_energy_drift








.. GENERATED FROM PYTHON SOURCE LINES 38-42

Pure vs. perturbed: the same manifest symmetry, a different hidden one
------------------------------------------------------------------------
Both systems share exactly the same central-force rotational symmetry;
only the perturbed one loses the extra structure special to :math:`1/r`.

.. GENERATED FROM PYTHON SOURCE LINES 42-78

.. code-block:: Python


    pure = KeplerSystem.from_orbital_elements(a=1.0, e=0.4)
    perturbed = KeplerSystem.from_orbital_elements(a=1.0, e=0.4, c_pn=0.01)

    res_pure = pure.integrate((0, 200), dt=1e-3, method="yoshida4")
    res_pert = perturbed.integrate((0, 200), dt=1e-3, method="yoshida4")

    E_pure, E_pert = relative_energy_drift(res_pure.energy), relative_energy_drift(res_pert.energy)
    L_pure, L_pert = angular_momentum_drift(res_pure.q, res_pure.p), angular_momentum_drift(res_pert.q, res_pert.p)
    A_pure, A_pert = lrl_drift(pure, res_pure.q, res_pure.p), lrl_drift(perturbed, res_pert.q, res_pert.p)

    print(f"pure 1/r:  max energy drift={E_pure.max():.2e}, max L drift={L_pure.max():.2e}, max |A| drift={A_pure.max():.2e}")
    print(f"perturbed: max energy drift={E_pert.max():.2e}, max L drift={L_pert.max():.2e}, max |A| drift={A_pert.max():.2e}")

    fig, axes = plt.subplots(1, 3, figsize=(13, 4.2), sharex=True)
    labels = (
        "time-translation\n$\\to$ energy",
        "rotation\n$\\to$ angular momentum",
        "hidden $SO(4)$\n$\\to$ Laplace-Runge-Lenz",
    )
    for ax, drift_pure, drift_pert, label in zip(axes, (E_pure, L_pure, A_pure), (E_pert, L_pert, A_pert), labels):
        # Each of these drifts is itself oscillatory and periodically crosses
        # exactly zero, so a plain semilogy plot would blow the axis out to
        # 300+ orders of magnitude at every crossing; floor at a fixed noise
        # level (well above float64 round-off, well below the LRL drift this
        # plot exists to show) to keep the comparison legible instead.
        ax.semilogy(res_pure.t, np.maximum(drift_pure, 1e-14), color="steelblue", label="pure 1/r")
        ax.semilogy(res_pert.t, np.maximum(drift_pert, 1e-14), color="firebrick", label="perturbed ($c_{pn} \\neq 0$)")
        ax.set_ylim(1e-14, 1.0)
        ax.set_title(label, fontsize=11)
        ax.set_xlabel("t")
    axes[0].set_ylabel("drift from t=0 value")
    axes[0].legend(fontsize=8)
    fig.suptitle("Only the hidden symmetry's conserved quantity notices the perturbation", fontsize=13)
    fig.tight_layout(rect=[0, 0, 1, 0.90])




.. image-sg:: /api/gallery/classical/newtonian/images/sphx_glr_plot_05_noether_hidden_symmetry_001.png
   :alt: Only the hidden symmetry's conserved quantity notices the perturbation, time-translation $\to$ energy, rotation $\to$ angular momentum, hidden $SO(4)$ $\to$ Laplace-Runge-Lenz
   :srcset: /api/gallery/classical/newtonian/images/sphx_glr_plot_05_noether_hidden_symmetry_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    pure 1/r:  max energy drift=5.71e-12, max L drift=1.24e-13, max |A| drift=6.12e-12
    perturbed: max energy drift=5.16e-12, max L drift=8.10e-14, max |A| drift=8.36e-02




.. GENERATED FROM PYTHON SOURCE LINES 79-85

Energy and angular momentum drift stay pinned at integrator round-off in
both columns -- time-translation and rotational symmetry are untouched by
a purely radial perturbation. Only the Laplace-Runge-Lenz panel tells the
two systems apart: flat for the pure potential, growing for the
perturbed one, tracking the same perihelion precession visualized
directly in :doc:`plot_02_kepler_precession`.

.. GENERATED FROM PYTHON SOURCE LINES 85-87

.. code-block:: Python


    plt.show()








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

   **Total running time of the script:** (0 minutes 2.099 seconds)


.. _sphx_glr_download_api_gallery_classical_newtonian_plot_05_noether_hidden_symmetry.py:

.. only:: html

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      :download:`Download Python source code: plot_05_noether_hidden_symmetry.py <plot_05_noether_hidden_symmetry.py>`

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