The correlation between Bertrand's theorem on classical closed orbits in a central potential and the occurrence of related dynamical symmetries is investigated.
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We are indebted to one of the reviewers who has pushed us to do it.
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Orbits in the harmonic case having the same shape as closed orbits in the Newtonian case (ellipses), we would expect more to find in the harmonic case an algebra similar to that provided by the LRL vector rather than an Abelian one. This last expectation relies also on the fact that there exists a transmutation between Newtonian forces and harmonic forces. See
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The condition of having stable circular orbits at any distance allows one to prove that the potential must be of the form
with
and
where β is a rational number, see Ref. 3, Sec. 3.6.
13.
After this article was completed, our attention was drawn to the work of
R. P.
Martinez-y-Romero
,
H. N.
Múñez-Yépez
, and
A. L.
Salas-Brito
, AIP Adv.
10
, 065003
(2020
), where the authors have developed ideas similar to ours although with a different approach.© 2021 Author(s). Published under an exclusive license by American Association of Physics Teachers.
2021
Author(s)
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