I. Launch
Dissipation chooses up.
The sphere loses mechanical energy at every slipping instant. The centre of mass can still rise because the final spin carries much less rotational energy. The conserved constraint is not energy. It is Jellett's quantity.
- inclination
- waiting
- height
- waiting
- energy change
- waiting
- Jellett drift
- waiting
- contact normal
- waiting
It climbs because energy is allowed to move between accounts.
The contact point slips. Friction takes mechanical energy away, but its torque also moves angular momentum between precession and spin. The higher configuration costs more gravitational potential energy and far less rotational energy. The total still falls.
The thread through the transfer is an exact integral for this contact geometry:
λ = -L · a = R I₁ φ̇ sin²θ - R I₃ ω₃(α - cosθ)
The animation evaluates that dot product from the current angular momentum vector and the current centre-to-contact vector. The plotted drift is numerical error, not a fitted correction.
In the angle trace, the rose inclination crosses the pale, numerically minimized effective-potential position as it climbs. That overlay is a numerical lens for the declared reference top. The narrower proof-backed family appears below.
II. The objection
Why upside down?
Friction explains why motion dies. It does not, by itself, explain which resting spin survives. The answer is a stability bifurcation: the low vertical state can lose stability while the inverted state gains it.
Group II · full inversion can be selected
The upright state loses stability above one spin condition and the inverted state gains it above another.
The strict single-minimum result is narrower than the trajectory model. The paper proves it for 1 - α² < γ < 1 together with a specific relation among mass, radius, and inertia. Inside that window this instrument enforces the relation and shows the proved rational potential. Outside it, the instrument refuses the potential claim and shows only the broader steady-state classification.
III. The check
An instrument with teeth.
The green lines are assertions, not captions. The offline verifier recomputes the page's numbers, takes an independent route through the published stability inequalities, and makes sure a deliberately damaged threshold is rejected.
What would fail? A wrong threshold breaks agreement with the independent upright and inverted stability inequalities. A wrong trajectory breaks conservation of Jellett's quantity, energy monotonicity, positive contact normal, or timestep convergence. A checker that accepts the poisoned threshold fails its own control.
The simulation also has a refusal path: if contact normal becomes nonpositive or the integration becomes nonfinite, it stops and reports the failed condition. The visible controls are bounded to the rectangle whose four corners the verifier accepts. It never silently clamps the force, angle, or contact.
What is free? The reference top is illustrative, not a commercial measurement. Its mass is 0.050 kg, radius 0.020 m, axial inertia 8.0e-6 kg m², eccentricity 0.30, inertia ratio 0.95, and viscous coefficient 0.30 s/m. Gravity is 9.81 m/s².
What is uncertain? Real contact includes deformation, dry friction, rolling resistance, and possible loss of contact. This page establishes what the stated viscous gliding model does. It does not claim every real top obeys that law, and it calls inversion entry into a neighbourhood rather than arrival at exactly π.
Apparatus
What this rests on.
- 01Rauch-Wojciechowski and Rutstam, Dynamics of an Inverting Tippe Top, 2014. Full gliding equations, threshold, Jellett integral, and the specially constrained effective-potential result.
- 02Ciocci and Langerock, Dynamics of the Tippe Top via Routhian Reduction, 2007. Independent vertical-state inequalities and the inertia-ratio classification.
- 03The complete reproducible method, caveats, and rerun command live in the repository research record for this page. The page itself makes no third-party request.