Question

A particle of mass m is attracted to a force center with the force of magnitude k/ r². Use plane polar coordinates and find Hamilton's equations of motion.

Answer 1

$F=\frac k{r^2},$

$L=T-U=\frac{m\dot r^2}2-Fr,$

$\frac{\partial L}{\partial r}-\frac d{dt}\frac{\partial L}{\partial \dot r}=0,$

$mr\dot r-\frac k{r^2}\ddot r=0.$