= Solution
In the equatorial plane,
$$
\dot r=p_r/m,\quad\dot\phi=p_\phi/(mr^2),\quad
\dot p_r=p_\phi^2/(mr^3)-k/r^2,\quad\dot p_\phi=0.
$$
Differentiating the given combination produces a cosine term
$$
-\frac{p_\phi}{m}\left(\frac{p_\phi^2}{mr^3}-\frac{k}{r^2}\right)\cos\phi
+\left(\frac{p_\phi^2}{mr}-k\right)\frac{p_\phi}{mr^2}\cos\phi=0
$$
and a <sine> term $p_rp_\phi\dot\phi\sin\phi/m-p_\phi^2\dot r\sin\phi/(mr^2)=0$. Thus $\boxed{\dot G=0}$. The expression is a Cartesian component of the <Laplace-Runge-Lenz vector> with a conventional overall <normalization>, reflecting the additional <symmetry> of the inverse-square-force problem.
Back to article page
