.. _sphx_glr_api_gallery_classical_rotations:

Rigid-body rotations
--------------------

Free and torqued rigid bodies: the intermediate axis theorem
(Dzhanibekov effect) for a torque-free top, rendered both abstractly
(the SO(3) momentum sphere) and literally (a tumbling 3D body); a heavy
symmetric top's precession and nutation under gravity; Euler's disk
settling into a finite-time singularity; and a rattleback's one-way
spin reversal.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="EulerTop is a torque-free rigid body: Euler&#x27;s equations for the body-frame angular velocity \boldsymbol\omega = (\omega_1, \omega_2, \omega_3) about the three principal axes, with moments of inertia I_1, I_2, I_3,">

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  .. image:: /api/gallery/classical/rotations/images/thumb/sphx_glr_plot_01_dzhanibekov_thumb.png
    :alt:

  :doc:`/api/gallery/classical/rotations/plot_01_dzhanibekov`

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      <div class="sphx-glr-thumbnail-title">The intermediate axis theorem (Dzhanibekov effect)</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="HeavySymmetricTop is a symmetric top with one point fixed, spinning under gravity, with generalized coordinates q = (\phi, \theta, \psi) -- the standard z-x-z Euler angles (precession, nutation, spin) -- and Lagrangian">

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  .. image:: /api/gallery/classical/rotations/images/thumb/sphx_glr_plot_02_heavy_symmetric_top_thumb.png
    :alt:

  :doc:`/api/gallery/classical/rotations/plot_02_heavy_symmetric_top`

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      <div class="sphx-glr-thumbnail-title">The heavy symmetric top: effective potential, precession and nutation</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The same torque-free rigid body as plot_01_dzhanibekov -- EulerTop, governed by Euler&#x27;s equations I_i\dot\omega_i = (I_j - I_k)\,\omega_j\omega_k (cyclic in i,j,k) for the body-frame angular velocity, coupled to the orientation quaternion. The SO(3) momentum-sphere picture (plot_01_dzhanibekov) shows the abstract geometry behind the intermediate axis theorem. This instead draws the actual rigid body -- a flattened &quot;paddle&quot; (book/phone-like) box, elongated along its smallest-moment axis and flattened along its largest-moment axis, exactly like the real object in the famous ISS video -- and rotates it in 3D with animate_rigid_body_tumble, which builds the wireframe box from half_extents and rotates its vertices at each frame via rotation_matrix, built from the integrated orientation quaternion. Spin it about the intermediate axis and it periodically flips end over end; spin it about either other principal axis and it just spins.">

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  .. image:: /api/gallery/classical/rotations/images/thumb/sphx_glr_plot_03_rigid_body_3d_thumb.gif
    :alt:

  :doc:`/api/gallery/classical/rotations/plot_03_rigid_body_3d`

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      <div class="sphx-glr-thumbnail-title">The Dzhanibekov effect, literally: a tumbling 3D rigid body</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="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 \theta (between the disk plane and the table) and precession angle \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), EulersDisk uses a reduced phenomenological pair of ODEs,">

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  .. image:: /api/gallery/classical/rotations/images/thumb/sphx_glr_plot_04_eulers_disk_thumb.gif
    :alt:

  :doc:`/api/gallery/classical/rotations/plot_04_eulers_disk`

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      <div class="sphx-glr-thumbnail-title">Euler's Disk: a finite-time singularity on a tabletop</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="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. Rattleback is a deliberately simplified toy model built directly on EulerTop&#x27;s free-rigid-body equations, for a reduced state (n_1, n_2, n_3) (n_3 playing the role of spin about the roughly-vertical axis, n_1, n_2 the two rocking/tipping-mode rates):">

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  .. image:: /api/gallery/classical/rotations/images/thumb/sphx_glr_plot_05_rattleback_thumb.gif
    :alt:

  :doc:`/api/gallery/classical/rotations/plot_05_rattleback`

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      <div class="sphx-glr-thumbnail-title">The Rattleback: one-way spin reversal</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/classical/rotations/plot_01_dzhanibekov
   /api/gallery/classical/rotations/plot_02_heavy_symmetric_top
   /api/gallery/classical/rotations/plot_03_rigid_body_3d
   /api/gallery/classical/rotations/plot_04_eulers_disk
   /api/gallery/classical/rotations/plot_05_rattleback