GYROSCOPE
Gravity turns the computed angular-momentum vector: full Euler–quaternion motion, exact regular precession, nutation, and an honestly bounded fast-top estimate.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Body and frame
A symmetric rigid body is pinned at one frictionless point. Its centre of mass lies along body axis e₃, tilted 30 degrees from lab vertical. A body-to-lab quaternion carries the complete orientation, so the rotor witness, symmetry axis, and every later vector have one unambiguous three-dimensional state.
Torque turns L
Gravity supplies τ = rCM × Mg. The lab angular momentum is R(q)Iω, not an arrow forced onto the symmetry axis. Euler’s three coupled body equations and quaternion kinematics evolve together; Lz and the body-axis component L₃ remain conserved even while gravity redirects L. Canvas vector lengths are direction-normalized for legibility; their directions and the displayed numerical magnitudes come from the shared state.
Exact branch, then estimate
The exact constant-tilt branch solves a quadratic, giving +0.465 rad/s. The familiar Mgr/(I₃ω₃) is shown separately as a fast-top estimate with a live δ test and a measured deviation; it is never relabeled exact.
Exact evolution
At 1× speed, six seconds of scene playback represents 13.5 physical seconds (2.25 times real-time). The projected top, CM, body witness, symmetry-axis path, ω, L, τ, Lz, four graph cursors, energy, and azimuth availability all sample the same integrated quaternion/Euler state. This initial condition is the exact slow regular branch.
Conservation audit
The endpoint is the actual final numerical state. Total energy, lab Lz, body L₃, quaternion norm, and a central-difference dL/dt = τ check report their worst residuals over the whole stored trajectory. |L| is deliberately not called conserved. The measured signed precession is compared directly with the conditional fast-top estimate.