.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/classical/rotations/plot_04_eulers_disk.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_api_gallery_classical_rotations_plot_04_eulers_disk.py: Euler's Disk: a finite-time singularity on a tabletop ========================================================== Spin a disk and set it rolling flat on a table and it settles into a shuddering near-steady state described, at each instant, by just its inclination angle :math:`\theta` (between the disk plane and the table) and precession angle :math:`\phi` (the azimuthal angle of the rolling contact point). Its contact point precesses faster and faster -- the characteristic rattle rising in pitch -- right up until the disk suddenly stops: a finite-time singularity. Rather than committing to one first-principles dissipation mechanism (viscous air drag vs. rolling friction are both argued for in the literature; see the history page), :class:`~physicskit.classical.systems.rotations.EulersDisk` uses a reduced phenomenological pair of ODEs, .. math:: \dot\theta = -\frac{\text{decay\_rate}}{\theta}, \qquad \dot\phi = \frac{\text{precession\_const}}{\theta} , in which both the precession rate and the (implied) dissipation rate diverge as :math:`\theta \to 0`, reproducing the same qualitative finite-time collapse. Ignoring the regularizing floor near :math:`\theta = 0`, the first equation integrates in closed form to :math:`\theta(t) = \sqrt{\theta_0^2 - 2\,\text{decay\_rate}\cdot t}`, vanishing at the finite collapse time :math:`t_f = \theta_0^2 / (2\,\text{decay\_rate})`. .. GENERATED FROM PYTHON SOURCE LINES 32-46 .. code-block:: Python import matplotlib.pyplot as plt from physicskit.classical.systems.rotations import EulersDisk, eulers_disk_theta_analytic from physicskit.classical.visualizers.animations import animate_eulers_disk theta0, decay_rate = 0.5, 0.05 t_f = theta0**2 / (2.0 * decay_rate) system = EulersDisk(theta0=theta0, decay_rate=decay_rate, precession_const=1.5) # Stop just short of the (regularized) collapse time -- theta keeps # decreasing past zero once floored, which is no longer physically # meaningful. result = system.integrate((0.0, 0.98 * t_f), dt=1e-1, method="implicit_midpoint") .. GENERATED FROM PYTHON SOURCE LINES 47-51 Animation: the contact point spiraling inward --------------------------------------------------- Watch the contact point spiral toward the center and precess faster and faster as theta collapses toward zero. .. GENERATED FROM PYTHON SOURCE LINES 51-55 .. code-block:: Python anim = animate_eulers_disk(system, result, interval=30, trail=600) plt.show() .. container:: sphx-glr-animation .. raw:: html

.. GENERATED FROM PYTHON SOURCE LINES 56-60 To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:: anim.save("eulers_disk_animation.gif", writer="pillow", fps=30) .. GENERATED FROM PYTHON SOURCE LINES 62-67 Numerical vs. analytic collapse ------------------------------------ The regularized numerical theta(t) tracks the exact, unregularized closed-form solution ``theta(t) = sqrt(theta0^2 - 2*decay_rate*t)`` almost all the way to the finite collapse time t_f. .. GENERATED FROM PYTHON SOURCE LINES 67-80 .. code-block:: Python theta_exact = eulers_disk_theta_analytic(result.t, theta0, decay_rate) fig, ax = plt.subplots(figsize=(8, 4)) ax.plot(result.t, result.y[:, 0], label="numerical (regularized)", color="steelblue") ax.plot(result.t, theta_exact, "k--", label="analytic (unregularized)") ax.axvline(t_f, color="gray", ls=":", label=r"$t_f$ (finite-time collapse)") ax.set_xlabel("t") ax.set_ylabel(r"$\theta$") ax.set_title("Euler's disk: finite-time collapse of theta(t)") ax.legend() fig.tight_layout() plt.show() .. image-sg:: /api/gallery/classical/rotations/images/sphx_glr_plot_04_eulers_disk_002.png :alt: Euler's disk: finite-time collapse of theta(t) :srcset: /api/gallery/classical/rotations/images/sphx_glr_plot_04_eulers_disk_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 1.517 seconds) .. _sphx_glr_download_api_gallery_classical_rotations_plot_04_eulers_disk.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_eulers_disk.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_04_eulers_disk.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_04_eulers_disk.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_