Form invariance of Newton's 2nd Law under Galilean Transformation

Question

I came across this question: https://www.quora.com/How-do-you-prove-that-Newtons-2nd-law-non-relativistic-takes-the-same-form-in-all-inertial-frames-under-Galilean-transformations

Proving that Newton's 2nd Law is form invariant under Galiliean transformations - and I've noticed the approach described in the answer to that question is pretty standard. Mainly we prove that the acceleration is invariant, so $\vec a=\vec a\ '$. The mass is assumed invariant - fair enough. But most treatments that I've looked at upon google go on to show that since $\vec F=m\vec a$, and since mass and acceleration are unchanged, therefore the force must be the same under Galilean transformations.

But this is a proof of the invariance of $\vec F$, and not of the 2nd Law itself. To prove the form invariance of the law, we also have to show that $\vec F=\vec F\ '$. So how do we justify that $\vec F=\vec F\ '$?

Answer 1

You can justify that $\vec{F} = \vec{F}'$ using Newton's second law which reads:

For any intertial frame of reference, $\vec{F} = m\vec{a}$.

Since accelaration is invariant under a Galilean transformation, then as you correctly said:

$$\vec{a} = \vec{a}'$$

Combining the above result with the Newton's second law implies that the force must be the same on both frame of reference, that is:

$$\vec{F} = m\vec{a} = m\vec{a}' = \vec{F}'$$

Answer 2

Galilean invariance is an additional physical principle not implied by Newton's 2nd law, which constrains $F$. It is very easy to imagine a force law which breaks Galilean invariance. We suppose there actually is a preferred rest frame, then a force like $F=\gamma v$, where $v$ is your velocity relative to the preferred frame, would break Galilean invariance.

Now, in fact you will find that drag forces are proportional to velocity. The reason these don't break Galilean invariance is that the drag force is really dependent on the relative velocity of the object and the fluid through which its moving. Galilean invariance is recovered if we boost both the fluid and the object. To break Galilean invariance, the preferred rest frame would have to be something intrinsic to the laws of physics themselves. There is no way to rule out this possibility from first principles, and there is research on theories which break Lorentz invariance, however no experiments have ruled out relativity, and proceeding under the assumption that it is true has been very fruitful in the development of physics.