Spin without a pivot. The inertia tensor turns torque into motion and maps angular velocity to angular momentum: L = Iω. With no external torque, L is fixed in space even as ω wobbles — the tensor, not a scalar, is what keeps the books. Rendered, not quoted.
A rigid body has a symmetric, positive-definite inertia tensor I. Its eigenvectors are the principal axes; its eigenvalues are the principal moments I₁,I₂,I₃. In that frame Euler’s torque-free equations govern ω:
Body used here: principal moments I = (2, 3, 4). Conserved: L = Iω, energy T = ½ωᵀIω, and ∥L∥.
Rotation in three dimensions. Euler’s rigid-body equations give the body a genuine orientation state — a rotation R in SO(3) evolving alongside ω. The inertia tensor is the bridge from angular velocity to angular momentum; conservation of L needs no pivot, only zero torque.
neighbour → the-angular-velocity · rotation as an evolving frame, det R = 1 throughout.
Live re-check of L-conservation along the true motion. Confirms while I is a tensor; flips red the instant the tensor is swapped for a scalar in window 6.
Torque-free spin started near the intermediate axis (I₂): ω₀ = (0.15, 6.0, 0.15). No external torque. Integrated with RK4 for ω and a Rodrigues exponential map for R (keeps R in SO(3)).
Spinning body (green) tumbles — the tennis-racket flip — while the angular-momentum vector L stays nailed in space.
Proven result: with a full inertia tensor and zero torque, the inertial angular momentum is conserved to integrator tolerance while ω wobbles freely.
wall “A spinning body has one number for its inertia, so L is just mass-times-spin and always lines up with ω.”
False for any body that is not a sphere. Off a principal axis, L and ω point in different directions; ω precesses around the fixed L. Real failure modes the tensor guards against: intermediate-axis instability (the Dzhanibekov flip), gyroscopic coupling, and treating a spacecraft or racket as an isotropic top.
ω is constant for a freely spinning body.
→ Only L and energy are conserved; ω is generally not — it wobbles, and about the intermediate axis it flips.
L always points along ω.
→ True only when ω lies on a principal axis or the body is isotropic; otherwise L = Iω tilts away from ω.
Energy conservation forces ω to hold steady.
→ Energy fixes ω to an ellipsoid; ∥L∥ fixes it to a sphere. Their intersection (polhode) lets ω roam.
Planted void (disclosed): replace the inertia tensor with one scalar moment Iₛ = (I₁+I₂+I₃)/3. Off-diagonal coupling and the distinct principal moments vanish, so L = Iₛω no longer stays fixed under free rotation. The Witness (7) catches it live.