.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/chaos/billiards/plot_circle_billiard.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_chaos_billiards_plot_circle_billiard.py: Circle Billiard (Integrable) ============================ A billiard is a point particle moving in a straight line at constant speed inside a closed boundary, until it strikes the boundary and undergoes specular reflection -- the outgoing velocity :math:`\mathbf{v}'` obtained from the incoming velocity :math:`\mathbf{v}` and the local unit normal :math:`\mathbf{n}` by .. math:: \mathbf{v}' = \mathbf{v} - 2(\mathbf{v}\cdot\mathbf{n})\,\mathbf{n}. The circular billiard, whose boundary is :math:`x^2 + y^2 = r^2`, is the simplest integrable 2D billiard: the angle of incidence at every bounce is exactly conserved, so the boundary phase-space coordinate :math:`\sin\phi` (the sine of the angle between the outgoing velocity and the local boundary tangent) never changes along a trajectory. This example plots a single trajectory and the resulting Poincare section. .. GENERATED FROM PYTHON SOURCE LINES 22-32 .. code-block:: Python import matplotlib.pyplot as plt import numpy as np from physicskit.chaos.systems.billiards import CircleBilliard from physicskit.chaos.visualizers.dynamic_plots import animate_billiard_trajectory from physicskit.chaos.visualizers.phase_space import plot_billiard_trajectory, plot_poincare_section billiard = CircleBilliard(radius=1.0) .. GENERATED FROM PYTHON SOURCE LINES 33-41 Launch off-center ------------------------ The circle's default interior point (used elsewhere for e.g. sampling starting positions) is its exact center -- and a ray launched from dead center hits the boundary head-on at *every* bounce, reflecting straight back through the center each time: a degenerate back-and-forth line, not the star-like pattern integrability is famous for. Any off-center point avoids this. .. GENERATED FROM PYTHON SOURCE LINES 41-43 .. code-block:: Python pos = (0.3, 0.0) .. GENERATED FROM PYTHON SOURCE LINES 44-48 Animation --------- Watch the ray bounce around the disk, tracing out the star-like pattern live alongside its Poincare section filling in point by point. .. GENERATED FROM PYTHON SOURCE LINES 48-52 .. code-block:: Python anim = animate_billiard_trajectory(billiard, pos=pos, vel=(0.3, 1.0), n_bounces=60, interval=50) plt.show() .. container:: sphx-glr-animation .. raw:: html

.. GENERATED FROM PYTHON SOURCE LINES 53-57 To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:: anim.save("circle_billiard_animation.gif", writer="pillow", fps=25) .. GENERATED FROM PYTHON SOURCE LINES 59-64 Trajectory ---------- A single ray, launched from an interior point, bounces around the disk forever tracing out a star-like pattern (unless its angle is a rational multiple of pi, in which case it eventually closes on itself). .. GENERATED FROM PYTHON SOURCE LINES 64-66 .. code-block:: Python fig, ax = plot_billiard_trajectory(billiard, pos=pos, vel=np.array([0.3, 1.0]), n_bounces=60) .. image-sg:: /api/gallery/chaos/billiards/images/sphx_glr_plot_circle_billiard_002.png :alt: CircleBilliard trajectory :srcset: /api/gallery/chaos/billiards/images/sphx_glr_plot_circle_billiard_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 67-76 Poincare section ---------------- Because the circle is integrable, ``sin(phi)`` is an exact invariant of the motion: each ray traces out its own horizontal line in the ``(s, sin(phi))`` section, at whatever ``sin(phi)`` its own impact parameter fixes, regardless of how many bounces are simulated -- together, many rays from the same off-center point foliate the section into a family of such lines (rays launched from dead center, by symmetry, would all share the same ``sin(phi) = 0`` and collapse onto a single line instead). .. GENERATED FROM PYTHON SOURCE LINES 76-79 .. code-block:: Python fig, ax = plot_poincare_section(billiard, n_rays=25, n_bounces=150, pos=pos) plt.show() .. image-sg:: /api/gallery/chaos/billiards/images/sphx_glr_plot_circle_billiard_003.png :alt: CircleBilliard Poincare section :srcset: /api/gallery/chaos/billiards/images/sphx_glr_plot_circle_billiard_003.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 5.451 seconds) .. _sphx_glr_download_api_gallery_chaos_billiards_plot_circle_billiard.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_circle_billiard.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_circle_billiard.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_circle_billiard.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_