In the post What symmetry causes the Runge-Lenz vector to be conserved?, and based on the results of https://arxiv.org/abs/1207.5001, it was it was discussed that the Runge-Lenz vector is the conserved charge of the on-shell symmetry $\delta\vec{r}=\vec{\epsilon}\times\vec{L}$. However, I don't understand the physical significance of this transformation. How could've someone guessed this symmetry before hand?
Some ideas:
- $\vec{L}\mapsto\vec{L}+\vec{\epsilon}\times\vec{L}$ is an infinitesimal rotation of $\vec{L}$. Does this help?
- The symmetry above is only a symmetry when the equations of motion are satisfied.
- Problem D of section 1.1. in P. Ramond, Field Theory: A Modern Primer. Westview Press, 1990. suggests that the interpretation has something to do with the fact that Newtonian orbits do not precess.
