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Inspired by this recent question, I would like to understand from a more general and mathematical perspective why closed orbits are only found for the Kepler ($V(r) \sim 1/r$) or harmonic ($V(r) \sim r^2$) potential problems, as follows from Bertrand's theorem.

There are two aspects that make these problems special, which I suspect may be related to the closed-orbit property. First, both problems are superintegrable. This property sits intuitively well with the idea that phase-space orbits should close "as quickly as possible", thus implying that real-space orbits close after a single revolution. Second, each problem possesses an additional "unexpected" conserved quantity, due to a larger symmetry of the problem than the obvious $O(3)$. For the Kepler problem, this is the Runge-Lenz vector, related to the $O(4)$ symmetry of the Hamiltonian. Meanwhile, the harmonic oscillator Hamiltonian conserves the Fradkin tensor: $$ F_{ij} = \frac{p_i p_j}{m\omega^2} + m\omega^2 q_i q_j, $$ which is related to an $SU(3)$ symmetry. In fact these symmetries and corresponding conserved quantities exist for any central field problem (D. M. Fradkin, Prog. Theor. Phys. 37 (1967), p.798). However the conserved quantities only take a "nice" form for the Kepler and harmonic problems, which also allows the corresponding quantum problems to be diagonalised exactly by symmetry arguments alone.

These considerations motivate the following question:

What specific physical/mathematical feature(s) do these two problems share that gives them the property of closed orbits? Does this feature bear relevance to the quantum counterpart?

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  • $\begingroup$ From the conservation of the Runge-Lenz vector, one may obtain Kepler orbits. $\endgroup$
    – Trimok
    Oct 1 '13 at 17:25
  • $\begingroup$ @Trimok Well this cannot be the full explanation, since all central potential problems possess a Runge-Lenz vector, as shown by Fradkin (see the link in my question). Indeed, Fradkin points out that the generalised Runge-Lenz vector and Fradkin tensors are actually associated with the fact that orbits can be restricted to one plane. However in general these generators are given by transcendental functions of the coordinates and momenta. $\endgroup$ Oct 1 '13 at 17:42
  • $\begingroup$ Besides integrable systems, KAM theorem shows that dynamical systems that are "close enough" to fully integrable have stable quasi-periodic orbits: scholarpedia.org/article/Kolmogorov-Arnold-Moser_theory $\endgroup$
    – Michael
    Oct 1 '13 at 17:51
  • $\begingroup$ Remember that these "orbits" are in phase space. Central potential problems can have closed phase-space orbits without having orbits that close in real space after a single revolution. $\endgroup$ Oct 1 '13 at 18:00
  • $\begingroup$ I wonder,if the specificity of Kepler/harmonic potential is not a high dynamical symmetry group as $SO(5,1)= SU(3,1)$. There is a discussion in Robert Gilmore, Lie Groups, Physics and Geometry, Cambridge, chapter 14, about hydrogenic atoms (and SO(4), SO(4,1), SO(4,2),SO(5,1)). Interestingly, relaxing the power-law interaction, for instance, with screened potentials, there exist closed orbits for suitable angular momentum values, see Ref. $\endgroup$
    – Trimok
    Oct 4 '13 at 10:54
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Very clear answer to the relation between the property that all bounded orbits are closed and superintegrability can be read off from the book of Landau (Mechanics) and Arnold ( Mathematical Methods of Classical Mechanics). In fact, Landau gives precise prescription for constructing superintegrals iff the orbits are closed. The main ingredients necessary to understand the problem are the Liouville tori, condition for the trajectory on the torus being closed and clear distinction between local and global integrals of motion.

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  • $\begingroup$ Which pages in Landau-Lifshitz and Arnold? $\endgroup$
    – Qmechanic
    Oct 7 '13 at 0:09
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    $\begingroup$ Last chapter of Landau-Lifshitz book where the properties of multidimensional motion are discussed from the point of view of action- angle variables. $\endgroup$
    – Kosinski
    Oct 31 '13 at 20:11
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    $\begingroup$ Landau-Lifshitz in the last paragraph $\S 52$ only discuss completely separable systems. $\endgroup$
    – Qmechanic
    Mar 27 '18 at 12:10
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This is not true that closed orbits exist only for harmonic and Keplerian potential. This is indeed true if we stay in the case of central potentials (see Bertrand's theorem). Otherwise, there are many other examples. See, e.g. "On higher symmetries in Quantum mechanics", Fris, Mandrosov, Smorodinsky, Uhlir and Winternitz.

$V(x,y)=a (x^2+y^2) +\frac{b}{x^2}+\frac{c}{y^2}$ will give all of its finite trajectories closed curves of degree 4. In case $b=c=0$ they will degenerate to ellipses.

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