Newton's law are form invariant under the coordinate substitutions:
$$ \tilde{x^{i}}=x^{i}+a^{i} $$
This means that Newtons' equation of motion,
$$ F^{i}=m \frac{d^{2} x^{i}}{d t^{2}} $$
(where $i=1,2,3)$ is the same in terms of the $\tilde{x}^{i}$ Is this the Gallielian Invariance ? How would I prove this ?
- If we define new set of coordinates as
$$ y^{i}=y^{i}\left(x^{k}\right) . $$
How would I find the equations of motion in these new cordinates assuming that $x^{k}$ are coordinates in an inertial frame? Which textbook should I be looking at ?