.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/classical/newtonian/plot_04_conservation_diagnostics.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_04_conservation_diagnostics.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_classical_newtonian_plot_04_conservation_diagnostics.py:


Angular momentum: a different conservation law from energy
================================================================

``physicskit.classical.utils.conservation`` has more than just energy-drift helpers.
``angular_momentum_2d`` and ``angular_momentum_drift`` track the planar
angular momentum

.. math::

    L = x p_y - y p_x ,

which is conserved whenever a system's dynamics are rotationally
symmetric about the origin (a central force, :math:`V = V(r)`) --
regardless of whether energy-conserving perturbations also happen to
break other symmetries. This contrasts two systems:

- KeplerSystem, even with a post-Newtonian perturbation that makes the
  Laplace-Runge-Lenz vector precess (a genuine physical effect: the
  perturbation breaks the *extra*, non-obvious SO(4) symmetry special
  to the exact 1/r potential), still exactly conserves angular
  momentum -- because it is still a central force, and central forces
  exert no torque about their center regardless of their radial profile.
- ProjectileMotion, a *uniform* gravitational field, is not a central
  force from the origin at all: gravity exerts a real torque there, so
  L changes -- growing steadily as the projectile falls.

.. GENERATED FROM PYTHON SOURCE LINES 30-35

.. code-block:: Python

    import matplotlib.pyplot as plt

    from physicskit.classical.systems.newtonian import KeplerSystem, ProjectileMotion
    from physicskit.classical.utils.conservation import angular_momentum_2d, angular_momentum_drift, relative_energy_drift








.. GENERATED FROM PYTHON SOURCE LINES 36-38

Kepler, with a precession-causing perturbation: L still conserved
------------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-45

.. code-block:: Python


    kepler = KeplerSystem.from_orbital_elements(a=1.0, e=0.5, c_pn=0.02)
    result_k = kepler.integrate((0, 100), dt=1e-3, method="yoshida4")
    L_drift_k = angular_momentum_drift(result_k.q, result_k.p)
    E_drift_k = relative_energy_drift(result_k.energy)
    print(f"perturbed Kepler: max |L(t)-L(0)| = {L_drift_k.max():.2e}, max relative energy drift = {E_drift_k.max():.2e}")





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

 .. code-block:: none

    perturbed Kepler: max |L(t)-L(0)| = 2.94e-14, max relative energy drift = 1.41e-11




.. GENERATED FROM PYTHON SOURCE LINES 46-48

Projectile motion: uniform gravity is not a central force, L drifts
--------------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 48-67

.. code-block:: Python


    proj = ProjectileMotion.from_launch(speed=20.0, angle_deg=45.0, g=9.81)
    result_p = proj.integrate((0, 2.9), dt=1e-3, method="yoshida4")
    L_p = angular_momentum_2d(result_p.q, result_p.p)
    print(f"projectile motion: L(0) = {L_p[0]:.3f}, L(end) = {L_p[-1]:.3f} (steadily changing, as expected)")

    fig, axes = plt.subplots(1, 2, figsize=(10, 4.3))
    axes[0].plot(result_k.t, L_drift_k, color="steelblue")
    axes[0].set_xlabel("t")
    axes[0].set_ylabel(r"$|L(t) - L(0)|$")
    axes[0].set_title("Perturbed KeplerSystem:\nangular momentum exactly conserved")

    axes[1].plot(result_p.t, L_p, color="firebrick")
    axes[1].set_xlabel("t")
    axes[1].set_ylabel("L(t)")
    axes[1].set_title("ProjectileMotion:\nnot a central force, so L is not conserved")
    fig.suptitle("Angular momentum conservation depends on rotational symmetry, not on energy conservation", fontsize=12)
    fig.tight_layout(rect=[0, 0, 1, 0.90])




.. image-sg:: /api/gallery/classical/newtonian/images/sphx_glr_plot_04_conservation_diagnostics_001.png
   :alt: Angular momentum conservation depends on rotational symmetry, not on energy conservation, Perturbed KeplerSystem: angular momentum exactly conserved, ProjectileMotion: not a central force, so L is not conserved
   :srcset: /api/gallery/classical/newtonian/images/sphx_glr_plot_04_conservation_diagnostics_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    projectile motion: L(0) = 0.000, L(end) = -583.378 (steadily changing, as expected)




.. GENERATED FROM PYTHON SOURCE LINES 68-72

This is the practical content of Noether's theorem: energy
conservation comes from time-translation symmetry, angular momentum
conservation from rotational symmetry, and the two can be switched on
or off independently, as shown here.

.. GENERATED FROM PYTHON SOURCE LINES 72-74

.. code-block:: Python


    plt.show()








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

   **Total running time of the script:** (0 minutes 0.663 seconds)


.. _sphx_glr_download_api_gallery_classical_newtonian_plot_04_conservation_diagnostics.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_04_conservation_diagnostics.ipynb <plot_04_conservation_diagnostics.ipynb>`

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

      :download:`Download Python source code: plot_04_conservation_diagnostics.py <plot_04_conservation_diagnostics.py>`

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

      :download:`Download zipped: plot_04_conservation_diagnostics.zip <plot_04_conservation_diagnostics.zip>`


.. only:: html

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

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