Duality of force laws and Conformal transformations

TL;DR

This paper explores a geometric duality between different classical force laws using conformal transformations, revealing relationships between physical constants and conserved quantities through differential geometry.

Contribution

It provides a simple derivation of the duality between force laws via conformal metric transformations and generalizes this result using differential geometry.

Findings

Mapping F~r and F~1/r^2 can be achieved by conformal transformations.

Derived relationships between G, M, E, and angular momentum for dual orbits.

Analyzed how conserved vectors transform under these dualities.

Abstract

As was first noted by Isaac Newton, the two most famous ellipses of classical mechanics, arising out of the force laws F~r and F~1/r^2, can be mapped onto each other by changing the location of center-of-force. What is perhaps less well known is that this mapping can also be achieved by the complex transformation, z -> z^2. We give a simple derivation of this result (and its generalization) by writing the Gaussian curvature in its "covariant" form, and then changing the \emph{metric} by a conformal transformation which "mimics" this mapping of the curves. The final result also yields a relationship between Newton's constant G, mass M of the central attracting body in Newton's law, the energy E of the Hooke's law orbit, and the angular momenta of the two orbits. We also indicate how the conserved Laplace-Runge-Lenz vector for the 1/r^2 force law transforms under this transformation, and compare it with the corresponding quantities for the linear force law. Our main aim is to present this duality in a geometric fashion, by introducing elementary notions from differential geometry.