Question proposed by my professor:

"Scaling: Consider a version of the Newtonian gravitational problem of n + 1 bodies where there is one large central mass at the origin of coordinates with mass M and n smaller masses in orbit with mass m. Think Saturn’s rings. Let G be the newtonian gravitational constant. I want you to consider scaling the system in such a way as to expose the only essential constant in this problem. Introduce new ”scaled” cartesian coordinates and time and thus momenta using scaling parameters Alpha and Beta. The Hamiltonian for this system in cartesian coordinates is:

SEE ATTACHMENT

Notice that in the above Hamiltonian, the nonlinear coupling terms are all in the third sum so you might expect the only remaining constant will be associated with these terms. Give a physical interpretation to your result."

Any help is appreciated. I don't know where to start. Thanks