Dimitri M. Christodoulou, Golden elliptical orbits in Newtonian gravitation,
Forum Geometricorum, 17 (2017) 465--471.
Abstract. In spherical symmetry with radial coordinate r, classical
Newtonian gravitation supports circular orbits and, for -1/r and
r^2 potentials only, closed elliptical orbits (J. Bertrand, M\'ecanique analytique,
C. R. Acad. Sci. Paris, 77 (1873) 849--853). Various
families of elliptical orbits can be thought of as arising from
the action of perturbations on corresponding circular orbits. We
show that one elliptical orbit in each family is singled out
because its focal length is equal to the radius of the
corresponding unperturbed circular orbit. The eccentricity of this
special orbit is related to the famous irrational number known as
the golden ratio. So inanimate Newtonian gravitation appears to
exhibit (but not prefer) the golden ratio which has been
previously identified mostly in settings within the animate world.
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