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The gravitational or Coulomb potential has a "hidden" symmetry (hidden in the sense that it does not follow from the rotational symmetry). The resulting integral of the motion (the Runge-Lenz vector) prevents space-filling orbits in classical mechanics (all orbits are closed), and introduces a degeneracy of the energy levels in quantum mechanics (energy ...

16

There are:
Bertrand’s theorem, which says that the isotropic oscillator and Kepler potentials are the only analytic radial ones all of whose nonrectilinear bounded orbits are closed. (Recommendation: Albouy - Lectures on the two-body problem. But he says: “The proof of this theorem provides very little “explanation” of the phenomenon.”)
The Levi-Civita–...

11

Physicists here. The input for a physical theory is always some topological space and some structure (such as a metric) that depends on the specific context. The dynamics are invariant under the isometries thereof. For example, the theory of Special Relativity deals with a manifold of the form $\mathbb R^n$, and with a (pseudo)metric $\operatorname{diag}(-1,+...

6

Here is an interpretation using symmetry reduction, but without explicitly using the Lenz-Runge vector (it's essentially an extended version of the example given in Cushman & Bates "Global aspects of classical integrable systems", p. 75).
Let $Q = \mathbb{R}^3$ be the configuration space (ignoring regularization issues at the origin) and let $P = T^* Q =...

4

The action-angle variables of the two-body graviational problem ('Kepler problem') are widely used in celestial mechanics community. These are called 'Delaunay variables' and make the toric structure of the phase space evident. See for example: Chang and Marsden - Geometric derivation of the Delaunay variables and geometric phases (CiteSeer published MSN) ...

1

There are some amazing aspects of hidden symmetry - Bertrand’s theorem connection. The first surprise is that it seems Runge-Lenz vector has relativistic (!) origin: https://www.sciencedirect.com/science/article/abs/pii/037596016891339X (Physical interpretation of the Runge-Lenz vector, by J.P.Dahl). The second surprise is that Runge-Lenz-like vectors do ...

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