Solution
= Solution
Let $\mu=GM_*$ and use plane polar coordinates in the orbital plane. The Newtonian <gravitational potential energy> is $U=-\mu M_p/r$, so
$$
\boxed{L=\frac{M_p}{2}(\dot r^2+r^2\dot\phi^2)+\frac{GM_*M_p}{r}.}
$$
A <central force> keeps the orbit in a plane. The star is treated as stationary, appropriate when $M_p\ll M_*$; otherwise the two-body reduction uses the <reduced mass> and the relative separation.