The Geometry of Isochrone Orbits: from Archimedes’ parabolae to Kepler’s third law. (arXiv:2003.13456v3 [physics.class-ph] UPDATED)
<a href="http://arxiv.org/find/physics/1/au:+Ramond_P/0/1/0/all/0/1">Paul Ramond</a>, <a href="http://arxiv.org/find/physics/1/au:+Perez_J/0/1/0/all/0/1">Jérôme Perez</a>
In classical mechanics, the Kepler potential and the Harmonic potential share
the following remarkable property: in either of these potentials, a bound test
particle orbits with a radial period that is independent of its angular
momentum. For this reason, the Kepler and Harmonic potentials are called
it{isochrone}. In this paper, we solve the following general problem: are
there any other isochrone potentials, and if so, what kind of orbits do they
contain? To answer these questions, we adopt a geometrical point of view
initiated by Michel H’enon in 1959, in order to explore and classify
exhaustively the set of isochrone potentials and isochrone orbits. In
particular, we provide a geometric generalization of Kepler’s third law, and
give a similar law for the apsidal angle, for any isochrone orbit. We also
relate the set of isochrone orbits to the set of parabolae in the plane under
linear transformations, and use this to derive an analytical parameterization
of any isochrone orbit. Along the way we compare our results to known ones,
pinpoint some interesting details of this mathematical physics problem, and
argue that our geometrical methods can be exported to more generic orbits in
potential theory.
In classical mechanics, the Kepler potential and the Harmonic potential share
the following remarkable property: in either of these potentials, a bound test
particle orbits with a radial period that is independent of its angular
momentum. For this reason, the Kepler and Harmonic potentials are called
it{isochrone}. In this paper, we solve the following general problem: are
there any other isochrone potentials, and if so, what kind of orbits do they
contain? To answer these questions, we adopt a geometrical point of view
initiated by Michel H’enon in 1959, in order to explore and classify
exhaustively the set of isochrone potentials and isochrone orbits. In
particular, we provide a geometric generalization of Kepler’s third law, and
give a similar law for the apsidal angle, for any isochrone orbit. We also
relate the set of isochrone orbits to the set of parabolae in the plane under
linear transformations, and use this to derive an analytical parameterization
of any isochrone orbit. Along the way we compare our results to known ones,
pinpoint some interesting details of this mathematical physics problem, and
argue that our geometrical methods can be exported to more generic orbits in
potential theory.
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