Noether's Ledger

Plate 4 speaks for itself: “Every symmetry keeps a promise.”

The Physics

Noether's theorem ties each continuous symmetry of the action to a conserved quantity. For a central potential the Lagrangian does not depend on the angle φ, so the angular momentum m r²φ̇ is conserved; Kepler's equal areas in equal times is that promise kept.

d/dt (m r² φ̇) = ∂L/∂φ = 0

• m: mass, kg
• r: radial distance, m
• φ: polar angle, rad
• φ̇: angular velocity, rad s⁻¹
• L: Lagrangian (1/2)m(ṙ² + r²φ̇²) − V(r), J
• V(r): central potential energy, J

L lacks φ. Equal areas in equal times.

Sources

• E. Noether, Nachr. Ges. Wiss. Göttingen, 235 (1918).

On the poster

Plate 004, “Noether's Ledger”: the pixel-art plate with its slogan “Every symmetry keeps a promise.”, the key equation and the facts panel shown above. Pixelated Physics. Physics. Personified.