Motion in a central field, the Kepler problem

Equipment

• Orbit model around a central mass with eccentricity slider e (0–0.95)
• Hamilton–Jacobi action surface S(q,t) drawn beside the orbit
• Rotatable canvas and parameter readout line
• Pause button to freeze the motion

Procedure

1. Watch the orbit as eccentricity changes

   Drag the e slider from 0 toward 0.95. At e = 0 the orbit is a circle; increasing e flattens the ellipse, with the central mass at one focus. Compare the orbit shape with $r(\phi) = p/(1 + e\cos\phi)$.

2. Gauge the angular speed along the orbit

   Press Pause at several positions and rotate the canvas for a better view. The body moves fast near periapsis and slow near apoapsis — the area law: $\frac{1}{2}r^2\dot\phi = \text{const}$, i.e. conserved angular momentum.

3. Compare with the action surface S(q,t)

   Look at the surface S(q,t) beside the orbit: its slope in q gives momentum $p = \partial S/\partial q$. Change e and predict how the surface deforms, then check on the figure; note this surface is one discretized illustrative solution, not the general action.

Simulation

Experiment history

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