Eccentricity and semimajor-axis invariant of a truncated resonance
= Eccentricity and semimajor-axis invariant of a truncated resonance
{title2=$e^2-\frac{q}{p+q}\log a=\mathrm{constant}$}
If retained <Lagrange planetary equations> have $\dot a=-2(p+q)ae^q C_r\sin\phi$ and $\dot e=-qe^{q-1}C_r\sin\phi$, then $d(e^2)/d\log a=q/(p+q)$. Thus $e^2-[q/(p+q)]\log(a/a_{\rm ref})$ is constant. A resonantly oscillating <orbital eccentricity>–<semi-major axis> trajectory retraces one curve rather than enclosing area in that projection.