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.