.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/classical/rotations/plot_05_rattleback.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_05_rattleback.py: The Rattleback: one-way spin reversal ========================================== A rattleback (Celt stone) spins smoothly forever in one sense, but spun the other way it soon wobbles, stalls, and reverses into the stable sense of spin. :class:`~physicskit.classical.systems.rotations.Rattleback` is a deliberately simplified toy model built directly on :class:`~physicskit.classical.systems.rotations.EulerTop`'s free-rigid-body equations, for a reduced state :math:`(n_1, n_2, n_3)` (:math:`n_3` playing the role of spin about the roughly-vertical axis, :math:`n_1, n_2` the two rocking/tipping-mode rates): .. math:: \dot n_1 = \frac{I_2 - I_3}{I_1} n_2 n_3 + \gamma n_3^2 - \mu n_1 - \eta n_1^3, \\ \dot n_2 = \frac{I_3 - I_1}{I_2} n_3 n_1 - \mu n_2 - \eta n_2^3, \\ \dot n_3 = \frac{I_1 - I_2}{I_3} n_1 n_2 - \mu n_3 , a damped Euler top plus a single term (:math:`\gamma n_3^2`, added to the :math:`n_1` equation only) that breaks the :math:`n_3 \to -n_3` symmetry an ordinary (non-reversing) Euler top has, standing in for the real rolling-contact-point physics, plus linear damping :math:`\mu` and cubic self-saturation :math:`\eta` so the instability it pumps up saturates and decays rather than diverging. .. GENERATED FROM PYTHON SOURCE LINES 33-39 .. code-block:: Python import matplotlib.pyplot as plt import numpy as np from physicskit.classical.systems.rotations import Rattleback from physicskit.classical.visualizers.animations import animate_rattleback .. GENERATED FROM PYTHON SOURCE LINES 40-44 A hard spin started predominantly about n3 ------------------------------------------------ With gamma > 0, spin started mostly in n3 (with only a tiny rocking seed in n1, n2) collapses, overshoots into reversed spin, and decays. .. GENERATED FROM PYTHON SOURCE LINES 44-48 .. code-block:: Python system = Rattleback(n0=[0.01, 0.01, 3.0], I1=1.0, I2=1.5, I3=2.0, gamma=1.0, mu=0.01, eta=0.01) result = system.integrate((0.0, 400.0), dt=0.1, method="rk4") .. GENERATED FROM PYTHON SOURCE LINES 49-53 Animation: spin direction and n3(t) side by side -------------------------------------------------------- The reversal is directly visible two ways: the indicator arrow reversing its sense of rotation, and the n3(t) trace crossing zero. .. GENERATED FROM PYTHON SOURCE LINES 53-57 .. code-block:: Python anim = animate_rattleback(system, result, interval=30, stride=20) plt.show() .. container:: sphx-glr-animation .. raw:: html

.. GENERATED FROM PYTHON SOURCE LINES 58-62 To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:: anim.save("rattleback_animation.gif", writer="pillow", fps=30) .. GENERATED FROM PYTHON SOURCE LINES 64-69 With gamma = 0: no reversal --------------------------------- Setting gamma to zero removes the symmetry-breaking term entirely, reducing the system to a damped Euler top that just spins down -- no reversal, regardless of which way it is spun. .. GENERATED FROM PYTHON SOURCE LINES 69-86 .. code-block:: Python system_no_reversal = Rattleback(n0=[0.01, 0.01, 3.0], I1=1.0, I2=1.5, I3=2.0, gamma=0.0, mu=0.01, eta=0.01) result_no_reversal = system_no_reversal.integrate((0.0, 400.0), dt=0.02, method="rk4") fig, ax = plt.subplots(figsize=(8, 4)) ax.plot(result.t, result.y[:, 2], label="gamma = 1.0 (reverses)", color="steelblue") ax.plot(result_no_reversal.t, result_no_reversal.y[:, 2], label="gamma = 0.0 (no reversal)", color="gray") ax.axhline(0.0, color="k", ls="--", lw=0.7) ax.set_xlabel("t") ax.set_ylabel(r"$n_3$ (spin rate)") ax.set_title("Rattleback: spin reversal requires the symmetry-breaking term") ax.legend() fig.tight_layout() plt.show() print(f"sign changes with gamma=1.0: {np.sum(np.diff(np.sign(result.y[:, 2])) != 0)}") print(f"sign changes with gamma=0.0: {np.sum(np.diff(np.sign(result_no_reversal.y[:, 2])) != 0)}") .. image-sg:: /api/gallery/classical/rotations/images/sphx_glr_plot_05_rattleback_002.png :alt: Rattleback: spin reversal requires the symmetry-breaking term :srcset: /api/gallery/classical/rotations/images/sphx_glr_plot_05_rattleback_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none sign changes with gamma=1.0: 69 sign changes with gamma=0.0: 0 .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 16.023 seconds) .. _sphx_glr_download_api_gallery_classical_rotations_plot_05_rattleback.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_05_rattleback.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_05_rattleback.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_05_rattleback.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_