The Heavy Top

Plate 2 speaks for itself: “Pulled down. Answers sideways.”

The Physics

A symmetric top spinning with one point fixed feels gravity as a torque that turns its axis sideways (precession) with a small nodding (nutation). In Euler angles the Lagrangian contains neither φ nor ψ, so the matching momenta are conserved.

L = (I₁/2)(θ̇² + φ̇² sin²θ) + (I₃/2)(ψ̇ + φ̇ cos θ)² − Mgℓ cos θ

• L: Lagrangian, J
• I₁, I₃: moments of inertia about a transverse axis through the fixed point and about the symmetry axis, kg m²
• θ, φ, ψ: Euler angles: tilt, precession, spin, rad
• M: mass, kg
• g: gravitational acceleration, m s⁻²
• ℓ: distance from fixed point to centre of mass, m

φ and ψ do not appear, so p_φ and p_ψ (kg m² s⁻¹) are conserved.

Sources

• H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 3rd ed., §5.7.

On the poster

Plate 002, “The Heavy Top”: the pixel-art plate with its slogan “Pulled down. Answers sideways.”, the key equation and the facts panel shown above. Pixelated Physics. Physics. Personified.