Spinning Top Precession: Why It Doesn't Fall

Launch

Initial spin rate ω₀ 1800 rpm

Mass distribution I₃/I₁ 1.80

Initial tilt θ₀ 18°

Friction / drag 0.045

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Spin ω₃

0 rpm

Precession Ω

0°/s

Tilt θ

0°

Stability ω₃/ω_crit

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How it works

This is a real symmetric top pivoting on a fixed point, integrated from the Lagrangian rigid-body equations (Euler angles θ, φ, ψ). Gravity's torque about the pivot does not act along the vertical (z) or the spin (symmetry) axis, so both the vertical angular momentum L_z and the spin component ω₃ would be exactly conserved for a frictionless top — that conservation is what forces the axis to precess around a cone instead of simply toppling:

φ̇ = (Lz − I₃ω₃cosθ) / (I₁sin²θ)
θ̈ = sinθcosθ·φ̇² − (I₃ω₃/I₁)·φ̇·sinθ + (Mgl/I₁)·sinθ
Ω_precession ≈ τ / (I₃ω₃),  τ = Mgl·sinθ

Friction/air drag at the tip bleeds spin energy (dω₃/dt = −k·ω₃) while Lz stays effectively fixed, so as ω₃ falls the precession rate Ω rises and the nutation term grows. Once ω₃ drops below the critical rate ω_crit = √(4I₁Mgl·cosθ)/I₃, the upright-spin equilibrium is no longer dynamically stable and θ runs away — the top visibly collapses onto the table, exactly as a real dreidel does at the end of its spin.

READY

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