Kepler–harmonic radial duality ($r=(\bar{E}/GM){\bar{r}}^2$)

Taking $\bar{E}=GMr/q^2>0$ in a radial orbit time transformation gives $r=(\bar{E}/GM)q^2$ and $\bar{\Phi }(q)=-E(\bar{E}/GM{)}^2{q}^2$. Bound Kepler orbits with $E<0$ map to a confining harmonic oscillator; positive energy maps to an inverted oscillator. Full angular reconstruction halves the original orbital angle.