Frame-dependent axial spin in a driven axisymmetric-body simulation

At one fixed parameter set, the reduced rotational model has two sharply different long-time outcomes. The ordinary basin settles to an 8.91° cone. A different basin sustains a broad, higher-energy angular motion whose θ range covers almost 147°.

The low-speed capture test starts from a phase on the established large branch, removes only the physical cylinder-axis component of angular velocity, resets the drive phase, and retains 7.1085 rad s⁻¹ of transverse motion. Over 200 s, the trajectory returns to the same statistical branch: mean physical cylinder-axis spin becomes −6.7837 rad s⁻¹ and mean total speed becomes 104.0241 rad s⁻¹. This is basin recapture from zero cylinder-axis spin, not spontaneous motion from complete mechanical rest.

The coordinate audit confirms the reported video observation. During the settled 120–200 s interval, the pink magnetic axis and the brown body +x marker each make approximately 80 net turns in the field direction. Both local axial-spin diagnostics remain negative at every sampled instant.

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The quantities answer different questions. ωbody,3 is the projection of the physical angular-velocity vector on the instantaneous magnetic body-z axis. The cylinder spin s = a·ω projects it on the physical cylinder axis a. Neither is the azimuthal rate of a painted marker in the laboratory.

For the settled benchmark, the first identity averages −2.506 rev s⁻¹, whereas the second averages +5.097 rev s⁻¹. Quaternion evolution already contains the moving-frame physics; body components are coordinates of the same angular-velocity vector in axes that rotate with the body. A visible marker combines precession, nutation, and local roll, so its laboratory azimuth can advance while local roll is negative.

For an axisymmetric cylinder with physical axis a, the inertia tensor may be written I = I⊥1 + (I∥−I⊥)aaᵀ. Projecting the damped Euler equation on that axis gives a particularly transparent balance:

If the magnetic axis and cylinder axis are exactly aligned, the ideal point-dipole torque is perpendicular to both and a·τB = 0; pure damping drives the axial spin toward zero. A manufacturing offset makes that projection nonzero. In the benchmark, approximately −6.7×10⁻⁸ N·m of mean cylinder-axis torque sustains the computed spin—approximately 0.032% of the simple mB sin δ upper scale.

Energy balance

The final 80 s ledger spans 4,800 field cycles. Field work supplies 118.822 μW; damping removes 118.921 μW; mechanical energy changes at −0.098 μW. The residual of the balance is 0.00042 μW. The small mechanical-energy decrease accounts for the field-to-damping difference.

This is a prescribed rotating-field model, so the drive is an energy source by construction. The important result is narrower: the spin is supported through the magnetic torque and not by the simulator’s optional psidd actuator.

Diagonalization of the supplied full inertia tensor identifies an axisymmetric cylinder with H/D = 0.51256 and a 2.1212° magnetic-to-physical-axis offset. This offset is 61.3% smaller than the 5.4762° benchmark value.

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The result remains statistically stationary and average-antiparallel: late mean θ = 102.764°, physical cylinder-axis spin = −8.195 rad s⁻¹, and U/mB = +0.1475 with 67.6% of samples on the A/P side. A standard angle/rate perturbation returns to the same statistics; LSODA and DOP853 agree to within 10⁻⁶° over the independent check.

The tilt azimuth also changed by about 90.2°, so Hamdi’s one file does not isolate tilt magnitude as the only varied parameter. The controlled ladder below removes that confound by reconstructing each full axisymmetric tensor at one fixed azimuth and holding the magnetic-frame launch, principal inertias, γ, mB, damping, drive, and psidd = 0 fixed.

Zero tilt closes the axial-torque channel—not the broad θ motion.

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At zero tilt, late θ still spans 147.155°, whereas physical cylinder-axis spin decreases from −0.6781 rad s⁻¹ to approximately −1.17×10⁻⁶ rad s⁻¹ at 200 s. Its maximum discrepancy from the analytic damping exponential is 1.26×10⁻¹³ rad s⁻¹. This result separates two mechanisms: manufacturing tilt is necessary for direct magnetic axial-spin exchange in the ideal point-dipole model, but not for broad θ motion from this launch.

• 200 s persistence. The benchmark’s late statistics are stationary under nonzero damping.
• Perturbation recovery. A deterministic attitude/rate perturbation returns to the same energy, θ span, and mean physical cylinder-axis spin.
• Cross-solver agreement. LSODA and DOP853 differ by less than 9.8×10⁻⁷° in θ during the strict 8 s comparison.
• Near-resonant torus. θ has a sharp 14.7493 Hz fundamental and harmonics, but the best tested 179-cycle full-state return remains far outside an exact-periodic closure gate.
• No chaos claim. A small shadow perturbation does not show positive finite-time divergence in the audit; quasiperiodic/near-resonant is the supported label.

A/P is also a statistical classification here, not an all-time constraint: late U/mB averages +0.1480 and is positive 67.15% of the time. The matched regular cone is strict A/P over its late window.

±0.10° of γ detuning does not create phase slips.

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This negative result shows that changing γ from 4.570079° to 4.47° or 4.67° does not visibly desynchronize the branch. Late θ spans remain within 146.815°–146.929°, and physical cylinder-axis spin remains between −6.923 and −6.642 rad s⁻¹. Because the rotating-frame phase is already continuously winding, counting its 11,794–11,799 wrap crossings as “phase slips” would be a category error.

A 0.396% kick difference selects opposite basins.

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The γ step returns statistically to baseline.

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Comparing the 10–30 s baseline with the restored 160–200 s interval, mean θ differs by 0.00684°, mean cylinder-axis spin by 0.000093 rad s⁻¹, and A/P occupancy by 0.000197. No irreversible hysteresis is observed for this schedule. This controlled step is not spontaneous decay, and a different microstate can still reach a different basin; the result demonstrates return for this trajectory rather than global reversibility.

Compact supporting records