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Scaling Hamiltonian
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

Re: Scaling Hamiltonian
Is this a Hamiltonian or a Lagrangian? Usually the Hamiltonian is equal to the total mechanical energy and I think that would require the two minus signs to be plus?
You could leave the spatial coordinates unchanged and set $\displaystyle t'=t+ \eta$ without changing the value of the Hamiltonian / Lagrangian. This results in conservation of energy.
The other common transformation in classical mechanics is:
$\displaystyle t'=t(1 + \eta)$
$\displaystyle x'=x(1 + \eta / 2)$
After you examine the units of G I expect that you would find this leaves the Hamiltonian/Lagrangian unchanged and would result in something being conserved.
What would you expect to be conserved? I would expect:
Total mechanical Energy
Linear momentum of the centre of mass
Angular momentum of the system