(a) Show that circular orbits exist for all attractive power-law central potentials. Find the…

(a) Show that circular orbits exist for all attractive power-law central potentials. Find the radius and total energy of the circular orbit as functions of the power of r and the angular momentum.

(b) Show that the orbits are stable only if  

(c) We mentioned, in preparation for the proof of Bertrand's theorem, that at aphelion (or perihelion)  = 0. But closure of the orbit requires also that  at aphelion (or perihelion) be the same at all aphelion (or perihelion) points. Prove that it is.

(d) Show that in central force motion the orbit is symmetric about  or about