REVIEW 1 cited by
A Novel Potential Featuring Off-Center Circular Orbits
Not yet reviewed by Pith; the record is open.
This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.
SPECIMEN: schema-true, not a live event
T0 review · schema-true
One-sentence machine reading of the paper's core claim.
pith:XXXXXXXX · record.json · timestamp
read the original abstract
In Book 1, Proposition 7, Problem 2 of his 1687 Philosophiae Naturalis Principia Mathematica, Isaac Newton poses and answers the following question: Let the orbit of a particle moving in a central force field be an off-center circle. How does the magnitude of the force depend on the position of the particle onthat circle? In this article, we identify a potential that can produce such a force, only at zero energy. We further map the zero-energy orbits in this potential to finite-energy free motion orbits on a sphere; such a duality is a particular instance of a general result by Goursat, from 1887. The map itself is an inverse stereographic projection, and this fact explains the circularity of the zero-energy orbits in the system of interest. Finally, we identify an additional integral of motion - an analogue of the Runge-Lenz vector in the Coulomb problem - that is responsible for the closeness of the zero-energy orbits in our problem.
Forward citations
Cited by 1 Pith paper
-
Hyperbolic Completion of Newton's Off-Center Orbit Problem: $SO(2,1)$ Symmetry, Inversion Duality, and Magnetic Classification
Zero-energy trajectories of V=−α/(R²−r²)² are arcs of Euclidean circles orthogonal to r=R (force center outside), with on-shell so(2,1) symmetry, inversion duality, and magnetic classification at Q²=8mαR².
Discussion (0). Continue with ORCID to comment.