I am not sure how your conservation of momentum (2) looks like, if an object could accelerate spontaneously then momentum would clearly not be conserved.

Hi jonny, I am not speaking about spontaneous acceleration. There are in general rectilinear and non-rectilinear forces. I just wonder whether it is possible or not for an isolated to be accelerated by utilizing non-rectilinear internal forces. According to my view, I think in order to answer this question it will be required a rigorous derivation (I haven't found any) of why an isolated system cannot accelerate through internal rectilinear forces.

get a book on general physics 101, it can be shown that the center of mass does not accelerate due to internal forces. The proof is the same as Dale's answer, but with a little more detail. It is not something that is unclear. Thus the answer is no, it will not accelerate if the third law is correct, even if forces were not on the same line.

This should help you, you can skip the irrelevant parts. link.springer.com/chapter/10.1007/978-3-030-15195-9_6 you want to understand 6.3.7 (note: a rigid body is just an instance of a system of particles)

@jonny I perfectly understand what is mentioned in 6.3.7 but that doesn't address the issue. Please see my last response to Dale (last comment at the bottom of this page). Dale's derivation is what one may find in classical mechanics literature that is conceptually wrong as I explained (see the last comment at the bottom of this page).

I dont really understand what you mean about the action being what moves the CM. What action force? but the link that I sent you does address your question, it is shown that the velocity of the center of mass is a constant if there are no external forces. That means the body will not accelerate. What step you do not understand? the demonstration does not assume that the forces are on the same line, only that $F_{21}=-F_{12}$

@jonny In other words, the net force due to internal forces has to be attributed just to the action forces. You cannot claim the net force equals to action and reaction. It is like saying a rocket (primary cause the mass the leaves the system causes its acceleration (reaction)) will never acquire a momentum.

"the net force due to internal forces is attributed just to the action forces" that does not make sense to me. Where in the demonstration did he assume that?

a rocket is an open system, particles leave the rocket. it does not apply here

you will have a better understanding if you read the link I posted, do not rely on Dale's short version

@jonny See his derivation, it is very clear as also you may ask Dale to confirm this. Again this is the misunderstanding I find in the literature when it is attempted to justify that an isolated system will never acquire momentum. Certainly, an isolated will never acquire momentum when the action-reaction is rectilinear but Dale's derivation (or what we read in today's literature) cannot rigorously confirm this.

I dont need to ask him, his answer is pretty clear

I am starting to suspect that you have not learn math or physics beyond high school, am I right? I m not asking to scold you, but to see why we fail to understand each other

@jonny I read the 6.3.7 and the author there makes the same mistake. The author takes as net force the sum of internal action and reaction which is not correct (again according to my understanding). Regarding rocketry, we have also there the action-reaction principle but the rocket acquires momentum. I didn't say it is the same as what we have in the isolated system. I pointed out that the principle (action-reaction) is the same that means the rocket moves because the net force equals the action force (inertial force -> mass that leaves the exhaust with a relative speed).

what other forces are acting over the body other than action-reaction pairs?

@jonny No other forces other than the internal action-reaction pair

then why you say it is wrong to consider only those forces?

you seem to suggest that not all of these forces should be included?

@jonny Because in our case (isolated system), the action applies to a part and the reaction to the rest of the system. You cannot put them (by definition of the action-reaction principle) as a net force that applies to the entire system. Therefore it would be similar by saying that the net force on a rocket equals action plus reaction resulting in no rocket acceleration which is wrong as you may understand. This is the whole point in this discussion.

@jonny said "you seem to suggest that not all of these forces should be included". They shouldn't be included as the net force that acts upon the entire system. It is not my suggestion but Newton's.

the center of mass of the rocket plus the propellant released into space does not accelerate (assuming the propellant does not interact with the environment). Regarding the other point: exactly, that is why internal forces doe not move the object, because action-reaction pairs cancel each other. The short reason is what you said, the proof is in the link

and you are just adding forces acting on every particle in the object. there nothing wrong with that. they are acting on different particles that belong to the object

@jonny said "the proof is in the link". Of course not! In order to see whether the isolated system should accelerate or not, one has to use the conservation of momentum. What one should expect is to address all the engaging momenta (as derived by the action-reaction principle) and prove that neither the part nor the system as a whole will ever change their momenta. It implies that the isolated system will never accelerate. This would be a rigorous derivation. There is no such derivation although trivial. Why so much noise about it? Because without it, none may ever answer my question.

You are right, got it. Brace for the Nobel prize!

@jonny "Brace for the Nobel prize!". Why do you say this? Isn't logical what I am asking for? How you may address my Question without knowing how to justify that there will be no change in momentum of the part and consequently of that of the entire system?

@jonny The key to prove that an isolated system can never accelerate is to find a way to demonstrate the momentum of a part will never change its momentum. Then, due to the action-reaction principle, we may rightly say the isolated system will never accelerate. It is straightforward and today's literature (even by going back to Newton's time) does not have an answer to this.