Dynamical symmetries behind Bertrand's theorem

J.

Bertrand

, “ ,” , – ().

V. I.

Arnold

, , 2nd ed. ( , ), Sec. 8.

H.

Goldstein

,

J. L.

Safko

, and

C. P.

Poole

, Jr. , , 3rd ed. ( , ).

L. D.

Landau

and

E. M.

Lifshitz

, , 2nd ed. ( , ).

This is in connection with the fact that the corresponding motions are completely degenerate (or commensurate); see Ref. 4, Sec. 50, Ref. 3, Sec. 10.7.

See, for example,

G. F.

Torres del Castillo

and

J. L.

Calvario Acócal

, , – ();

M.

Cariglia

and

E. S.

Araújo

, , – (); and references therein.

P. S.

Laplace

, ( , ), Tome I, Première partie, Livre II, p. and following;

C.

Runge

, ( , , );

W.

Lenz

, “ ,” , – ().

See, for example, Ref. 3, Sec. 9.8;

J.

Sivardière

, , – (); and references therein.

We are indebted to one of the reviewers who has pushed us to do it.

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

T.

Needham

, , – ();

Daniel R.

Stump

, , – ().

The action of the LRL vector as a Lie transformation or as a canonical transformation has already been investigated, see

H. H.

Rogers

, , – ();

T.

Ligon

and

M.

Schaaf

, , – ();

G. E.

Prince

and

C. J.

Eliezer

, , – ();

J.

Krause

, , – (); and references therein.

The condition of having stable circular orbits at any distance allows one to prove that the potential must be of the form $− K / r α$ with $K α > 0$ and $α = 2 − β 2$ where β is a rational number, see Ref. 3, Sec. 3.6.

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

, , (), where the authors have developed ideas similar to ours although with a different approach.