Thanks so much for your comments, I'm sorry for the late response. For some reason I didn't receive any notification of your message on Facebook...but anyway...here goes.
"You write two Ls for independent systems then define some relationship between the systems (in your case 2) then try and solve for the Hamiltonian of one system in terms of the other."
I think that if we were to write a single Lagrangian with the potential GMm/r, we are effectively assuming an interaction between two particles. What I mean is that this potential can be read as GMm/r = mv^2. A physical intepretation of this is that particle M gives, or more precisely assigns, a velocity v to particle m. The potential is itself the result of the interaction between the two particles. Best illustrated if M = 0. The potential is an interaction term. When we talk of Lagrangians I think that we cease to speak of independent particles or systems. We find it probably more appropriate to speak of single pairs of systems.
A single particle/wave/uniform field on its own at infinity does not have a velocity to speak of, nor even a coordinate system of its own. (When I say no coordinate system, I mean none that it can identify. The idea of a particle at infinity is defined by the observer who imagines that there is a particle at infinity, not by the particle itself.)
" This approach assume apriori knowledge of the system dynamics and will not result in meaningful physics in general."
I agree, in the sense that there is a problem with using Lagrangians, higlighted here, because the mass M of the observer must be measured in order to determine the size of the potential of particle m. But this is the nature of gravity, at least Newtonian gravity (though I think the potential g_ab(x) has the same problem/solution) and perhaps all other fields...if you measure the value of the classical field at one point then this provides information for the values of the field at other points.
So there is in a sense apriori knowledge that can be obtained from measuring classical ( and quantum?) fields. But the physics is meaningful. In other words, even at a quantum level, aspects of this method of extrapolition persist, if gravity is accurately described using fields.
The mass M is usually defined as a system of particles that can perform a measurement, the measuring device, on a particle m.
"A more consistent approach includes defining the system to include both particles. The Lagrangian will now include (if you desire) a term that addresses the interaction between the two systems. Now solve the system accordingly."
If we consider the above, that the potential in the Lagrangian contains information about an external observer, then we immediately realise that the potential is an appropriate term. For example, a change in the value of M reflects as a change in the potential of m, according to M. Before we start imagining problems relating to instantenous/action at a distance ideas, we must realise that the Lagrangian is written by the observer - not by the particle itself.
" In quantum field theory the systems to which I refer are called fields and they are not discrete entities as in your example but are present throughout spacetime. Additionally, take care when you say that "Spacetime is as a consequence of observation". A more accurate statement would be to say that human observation make our understanding of spacetime possible."
Up to this point we haven't started addressing the issue of quantization. We haven't even properly addressed the four-component forms and suitable tensors which are to be quantized. We have found an interpretation and used that to discuss the Lagrangian.
"Spacetime is as a consequence of observation". I couldn't find where I said this? If I said it, I'm obviously mistaken or have been presumptious.
"A more accurate statement would be to say that human observation make our understanding of spacetime possible." I agree.
