hilbert2 said:
Even if the laws of nature were something different than what they are, it would always be possible to define a force ##F## by saying that ##F = ma## is what it means.
While you could define force in this way, the result would be that the laws of motion would depend on the inertial frame of reference one chooses.
So, a mass being pulled by a spring stretched a constant amount would not experience constant acceleration i.e. where there is no observable physical change in the source of whatever it is that is causing the change in motion of a body, the change in motion would not be constant with time.
Consequently, a body M of mass m in IFR1 at rest (an inertial frame which defines a rest state) subjected to such a spring would experience a change in velocity of δv
1 in time δt and a different change of velocity of δv
2 over a subsequent equal time interval δt.
Now we try the first part of the experiment in IFR2 traveling at δv
1 relative to IFR1. We would do the first part of the identical experiment on an identical body M' of mass m commencing after the first δt interval.
Question: would we observe:
1. a change in the velocity of M' of δv
1 over the time interval δt? or
2. a change in velocity of M' of δv
2 over that time interval δt?
If the frames of reference IFR1 and IFR2 were equivalent, since the experiment in IFR2 is identical to the first part of the experiment in IFR1, we should see a change of velocity of M' in IFR2 of δv
1.
But the problem is that IFR2 is identical to the frame of reference of the body M after having its velocity changed by δv
1. And the change in velocity of M in the second interval δt is δv
2, not δv
1.
So, we must conclude that the inertial frames of M after δt and IFR1 are not equivalent, despite there being no distinguishing feature between them.
Therefore, the laws of physics would depend on the inertial frame of reference one chooses.
In other words, if all inertial frames measure time the same and if the laws of motion are the same in all inertial frames of reference, an unchanging agent acting effecting a change in motion of a body of a given mass will produce equal changes in velocity in equal times. And that is why F=ma is more than just an arbitrary definition.
AM