The conformal map w (z)\fc wzw(z) transforms the trajectory with energy −B-B−B in potential U (z)≔A∣dw/dz∣2\fc Uz\ceq A\v{\d w/\d z}^2U(z):=A∣dw/dz∣2 into the trajectory with energy −A-A−A in potential V (w)≔B∣dz/dw∣2\fc Vw\ceq B\v{\d z/\d w}^2V(w):=B∣dz/dw∣2. I will prove this beautiful result and show some implications of it.
https://ulysseszh.github.io/physics/2023/12/22/conformal-trajectory.html
The conformal map w (z)\fc wzw(z) transforms the trajectory with energy −B-B−B in potential U (z)≔A∣dw/dz∣2\fc Uz\ceq A\v{\d w/\d z}^2U(z):=A∣dw/dz∣2 into the trajectory with energy −A-A−A in potential V (w)≔B∣dz/dw∣2\fc Vw\ceq B\v{\d z/\d w}^2V(w):=B∣dz/dw∣2. I will prove this beautiful result and show some implications of it.