I do appreciate you trying to help me, Elfmotat. The background to all this is somewhat involved. What follows is a trade off between brevity and coherence. I aim to meet you somewhere in the middle. Okay here goes...
The project concerns the principle of mass additivity, which states that the mass of a composite object is the sum of the masses of its elementary parts. In most textbook discussions of Newtonian mechanics, this principle is simply presupposed, despite playing important roles (e.g. in the proof that centres of mass obey Newton's laws). However, some textbooks are more rigorous and set out to prove mass additivity, to show that "mass additivity is a consequence of Newtonian laws". I have attached an example (taken from Kibble & Berkshire's 'Classical Mechanics' (p12)).
The proof is only ever given in the context of inertial mass, and involves something like the following move. Where ##F_T## is the total force on a number of point masses indexed by i (due to an external source, which I do not specify, so as to simplify the equations), we begin with:
\sum_i m_i\mathbf{a}_i = \mathbf{F}_T
It is then assumed that the particles indexed by i compose a composite body and that the force on the composite is equivalent to the total force on the parts (##F_T=F_c##). A physical situation is then assumed in which the accelerations of the particles indexed by i are identical, which allows the following transformation:
\mathbf{a}_i\sum_i m_i = \mathbf{F}_c
It is then assumed that the acceleration of the composite ##\mathbf{a}_c## just is the acceleration of its parts so that:
\mathbf{a}_c\sum_i m_i = \mathbf{F}_c
Which is taken to show that given the force and acceleration of the composite, the composite's mass must be additive, that is:
\sum_i m_i = m_c
Hence mass is additive.
There is an objection to this proof, which focuses on the simplification that the acceleration of the parts are all identical. One way to show that this simplification is problematic is to show that the analogous simplification required to prove gravitational mass additivity is unphysical. The analogous simplification requires, not identical accelerations, but identical positions. The proof starts with the fundamental gravitation law:
\mathbf{g}(\mathbf{x})=G\sum_i \frac{\mathbf{x}-\mathbf{x}_i}{|\mathbf{x}-\mathbf{x}_i|^3}~m_i~
Now the analogue of the acceleration simplification: the positions of the particles indexed by i are identical, so that the following transformation is possible:
\mathbf{g}(\mathbf{x})=G \frac{\mathbf{x}-\mathbf{x}_1}{|\mathbf{x}-\mathbf{x}_1|^3}\sum_i~m_i~
Assuming that the composite determines the same field as the parts (##\mathbf{g}(\mathbf{x})=\mathbf{g}_c(\mathbf{x})##) and that the composite is positioned where its parts are positioned, we can derive gravitational mass additivity in the same way. But, so the objection goes, the proof fails because it assumes the unphysical assumption of particle position overlap. This enables one to object to the inertial proof with the following argument:
(1) If inertial mass additivity is a consequence of Newton's inertial laws then gravitational mass additivity is a consequence of Newton's gravitation law.
(2) It is not the case that gravitational mass additivity is a consequence of Newton's gravitation law. (From above.)
(3) Therefore, it is not the case that inertial mass additivity is a consequence of Newton's inertial laws.
Okay, so we've seen (i) a standard textbook proof for mass additivity and (ii) an objection to that proof. There are surely many ways of responding to the objection. I am looking at one type of response. This response states that the simplifications are only a product of the simple formalism chosen in the proofs. The response sets out to show that if we can formulate the equations so that they can cope with composites whose parts have varying positions and accelerations then a more general proof is possible. Enter Dirac delta functions.
The gravitational proof can be formulated as follows:
Fundamental many-particle law:
\mathbf{g}(\mathbf{x})=G \sum_i\int \frac{\mathbf{x}-\mathbf{r}}{|\mathbf{x}-\mathbf{r}|^3}~m_i~\delta(\mathbf{r}-\mathbf{x}_i)~\mathrm{d}^3\mathbf{r}
Recover the form of the single particle law:
\mathbf{g}(\mathbf{x})=G\int \frac{\mathbf{x}-\mathbf{r}}{|\mathbf{x}-\mathbf{r}|^3}\sum_i ~m_i~\delta(\mathbf{r}-\mathbf{x}_i)~\mathrm{d}^3\mathbf{r}
Assume that the field determined by the composite is the field determined by its parts:
\mathbf{g}_c(\mathbf{x})=G\int \frac{\mathbf{x}-\mathbf{r}}{|\mathbf{x}-\mathbf{r}|^3}\sum_i ~m_i~\delta(\mathbf{r}-\mathbf{x}_i)~\mathrm{d}^3\mathbf{r}
Infer gravitational mass additivity.
Then in the inertial case:
Fundamental many-particle law:
\sum_i \int m_i(\mathbf{x}) \mathbf{a}_i(\mathbf{x})\delta (\mathbf{x}-\mathbf{x}_i)d^3\mathbf{x} = \mathbf{F}_T
Recover the form of the single particle law (this involves deriving ##\mathbf{a}_c(\mathbf{x})##, the acceleration distribution of the composite, from the acceleration distributions of the parts, which I'll just assume here):
\int \mathbf{a}_c(\mathbf{x})\sum_i m_i(\mathbf{x}) \delta (\mathbf{x}-\mathbf{x}_i)d^3\mathbf{x} = \mathbf{F}_T
Assume that the force on the composite is the force on the parts:
\int \mathbf{a}_c(\mathbf{x})\sum_i m_i(\mathbf{x}) \delta (\mathbf{x}-\mathbf{x}_i)d^3\mathbf{x} = \mathbf{F}_c
Infer inertial mass additivity.
Sorry about the length of the post. I suspect people are having trouble with my inertial laws because I'm not defining the external force that determines the acceleration of all the parts? I thought leaving the external force undefined simplifies the equations, but that may be the thing that's causing problems.
Anyway, I'll stop now. I am very interested in your thoughts on this!
