# Symmetry considerations of linear momentum conservation

I was watching a video describing a way to deduce Newton's Third law using symmetry considerations.
Consider a system of $$N$$ particles not acted upon by any external force (isolated system). Suppose the medium in which the system in placed is homogenous.
Now we are displacing each particle in the system simultaneously and in the same direction such that the configuration of the system remains the same.
Now it has been said that the displacement which we have done is instantaneous i.e., there is no time interval involved in making such displacement. So it is just a thought process. Such displacements are known as virtual dislacements. So, there is no question of what agency has brought out this displacement.
Now we are trying to find out the virtual work done by the internal forces (as external force = $$0$$)
It has been said that $$\text{Virtual work done}\;=\;0\tag{1}$$
So, $$(\sum_i\vec F_i).\vec{\delta s}=0$$
As the above equation holds for all $$\delta\vec s$$
By Newton's second law,
$$\frac{d}{dt}\sum_i\vec{p_i}=0\implies\frac{d}{dt}\vec P=0$$
So, total momentum is conserved.
If we have two particle system, then
$$\sum_{i=1}^2\vec F_i=0\implies \vec{F_1}+\vec{F_2}=0$$
Thus, proving Newton's Third Law.

I have a doubt that the proof relies on $$(1)$$. I am not able to understand how $$(1)$$ is true?
In Goldstein's Classical Mechanics, momentum conservation has been proved through virtual work but by considering $$(1)$$ as a consequence of Newton's Third Law.
But here, I am not able to understand how we are considering $$(1)$$ to be true? I am not abe to understand that.

• Similar questions have been asked here before several times, check out for example physics.stackexchange.com/questions/12122/…. It isn't strictly enough to assume that translation (what you called "virtual displacement") doesn't change the forces. You also need to assume that Lagrangian stationary-action is a principle of the universe, and other stuff (like only two-body interactions, rotational invariance, etc.) to strictly reach the third law. May 8, 2023 at 9:22