The Dzhanibekov effect, literally: a tumbling 3D rigid body

The same torque-free rigid body as The intermediate axis theorem (Dzhanibekov effect) – EulerTop, governed by Euler’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 (The intermediate axis theorem (Dzhanibekov effect)) shows the abstract geometry behind the intermediate axis theorem. This instead draws the actual rigid body – a flattened “paddle” (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.

import matplotlib.pyplot as plt

from physicskit.classical.systems.rotations import EulerTop
from physicskit.classical.visualizers.animations import animate_rigid_body_tumble

# Half-extents (a, b, c) along body axes (1, 2, 3): a > b > c makes
# I1 = b^2+c^2 (smallest) < I2 = a^2+c^2 < I3 = a^2+b^2 (largest),
# matching the EulerTop's I1=1, I2=2, I3=3 below.
HALF_EXTENTS = (1.5, 1.0, 0.4)

Spin about the intermediate axis: it periodically flips end over end

top = EulerTop(omega0=[0.01, 1.0, 0.01], I1=1.0, I2=2.0, I3=3.0)
result = top.integrate((0, 30.0), dt=1e-3, method="implicit_midpoint")

anim = animate_rigid_body_tumble(top, result, half_extents=HALF_EXTENTS, interval=50, stride=200)

plt.show()

To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:

anim.save("rigid_body_tumble_animation.gif", writer="pillow", fps=30)