# Criteria on good inertial system approximation

I'm currently wrapping my head around Newton's First Law. I think I start to get a basic understanding on the meaning of this law in terms of "the existence of inertial system".

Basically my understanding is that the existence is an assertion based on experiment result (with Earth ground in mind), and there's no intrinsic reason for such a thing exist. I imagine there should be cases where Earth ground fails to become a good approximation of inertial frame, which leads to my question:

Within Newtonian mechanics where inertial frame is only an approximation, are there any guidelines on when it's appropriate to view some object as inertial reference frame so that the physical law would be mostly the same as Newtonian mechanics? I'm talking about guidelines like object relative size, distance, etc.

## 1 Answer

This is an interesting question! I apologize if the following is unnecesarily verbose, but there's a lot of math to keep track of.

If you have an object $$P$$ moving around in an arbitrary time-dependent (but rigid) coordinate frame $$\mathcal{B}$$ with possibly moving origin $$\mathcal{O_B}$$, you can check how accurately Newton's laws describe the dynamics of that object as it if it was in an inertial reference frame by checking the magnitude of the acceleration components not present in an inertial frame relative to some arbitrary stationary frame $$\mathcal{F}$$ with origin $$\mathcal{O}$$.

Concretely, you can calculate some sort of inertial ratio vector $$\vec{\mathcal{I}}$$ that compares the magnitudes of the accelerations you'd see if $$P$$ was moving in an inertial frame versus the accelerations you'd see if it was not:

$$\vec{\mathcal{I}} = \frac{\vec{a}_{P/\mathcal{B}}}{\vec{a}_{\mathcal{O}/\mathcal{O'}}+ 2\vec{\omega}_\mathcal{B}\times\vec{v}_{P/\mathcal{B}}+ \vec{\alpha}_\mathcal{B}\times\vec{r}_{P/\mathcal{O'}}+\vec{\omega}_\mathcal{B}\times(\vec{\omega}_\mathcal{B}\times\vec{r}_{P/\mathcal{O'}})+\vec{a}_{P/\mathcal{B}}}$$

where $$/$$ indicates "relative to", $$\vec{r}$$'s, $$\vec{v}$$'s and $$\vec{a}$$'s are positions, velocities and accelerations respectively, and $$\vec{\omega}$$'s and $$\vec{\alpha}$$'s are angular velocities and accelerations respectively.

If $$\mathcal{B}$$ is inertial, then $$\vec{\mathcal{I}} = \vec{1}$$; the non-inertial effects can bring this down to at worst $$\vec{\mathcal{I}} = \vec{0}$$, which is also the case if the object $$P$$ is not accelerating relative to $$\mathcal{B}$$ (i.e. there are no net "real" forces on the object $$P$$ in this frame).

Hope this helps!