Physics – Classical Physics
Scientific paper
2010-10-11
Am. J. Phys. 79: 6,2011
Physics
Classical Physics
8 pages, 1 figure; typos fixed, references added, matches version accepted in Am. J. Phys
Scientific paper
10.1119/1.3553231
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.
No associations
LandOfFree
Duality of force laws and Conformal transformations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Duality of force laws and Conformal transformations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Duality of force laws and Conformal transformations will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-607141