Central-Force Orbit Gallery
A particle in a central potential $V(r)=k r^p$, integrated by symplectic velocity-Verlet. The orbit is the primary scene; a side panel shows the effective potential $V_{\text{eff}}(r)=V(r)+\frac{L^2}{2\mu r^2}$ with the energy line and radial turning points. Presets walk through the Bertrand closed orbits (Kepler ellipse, harmonic oscillator), a precessing rosette and an unbound escape.
WHAT TO TRY
- Leave the exponent at $p = -1$ (gravity): the orbit is a closed ellipse that retraces itself forever.
- Nudge $p$ away from $-1$ and the ellipse stops closing, sweeping out a slowly turning rosette, a flower. Only gravity ($1/r^2$) and the spring ($r$) close.
- In the lower panel, the energy line cuts the effective-potential well at the turning points (red): raise the angular momentum $L$ and the well lifts, squeezing the orbit.