There is an extra conserved quantity, the Runge-Lenz vector. Look up the excellent Wiki page on this topic. -- Gene On Tuesday, January 16, 2018, 2:37:11 PM PST, Cris Moore <moore@santafe.edu> wrote: if I recall correctly, for 1/r^2 gravity there is a conserved quantity other than the energy and the angular momentum — this is why this system is “integrable". this additional quantity puts a bunch of elliptical orbits into an equivalence class with the same period. but I don’t remember how it works… can someone tell us? - Cris
On Jan 16, 2018, at 1:02 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Kepler's [third?] law, which states that the squared period is proportional to the cubed orbital radius, is derivable from dimensional analysis.
Of course, the dimensional analysis 'proof' of Kepler's third law is somewhat dissatisfying to me because it doesn't say *which* radius should be cubed -- this matters if different planets have different eccentricities.
[It happens that it's the semi-major axis, but this is by no means obvious. Does anyone know an *elegant* justification for this being the case (i.e. not involving churning through lots of coordinate calculations)?]
Best wishes,
Adam P. Goucher