By the Galilean transformations, one can easily derive that two different inertial observers 1,2 always measure the same forces. That is:

$$ \vec{F_1} \ \left(\vec{r_1}, \frac{d\vec{r_1}}{dt_1},t_1\right) = \vec{F_2} \ \left(\vec{r_2}, \frac{d\vec{r_2}}{dt_2},t_2\right) \ \ \ \ (1)$$

with $\vec{r_1}$ and $\vec{r_2} $ satisfying:

$$ \vec{r_1} = \vec{r_0} + t\vec{v_0} \ + \vec{r_2} \ \ \forall \ t\in \mathbb R$$

and $ t_1 $ and $ t_2 $: $$ t_1 = t_2 + a $$ (All the constants have the usual meaning)

Are there any force fields that violate (1) ? and how can one prove this ?


Yes. Electromagnetic forces are not invariant under Galilean transformations. (Because they transform instead under Lorentz transformations.)

For a simple example, consider two point charges $q$ travelling side by side at speed $v$ in the same direction, distance $d$ apart. The attractive magnetic force between them is


Now do a Galilean transformation into a frame moving along with the two particles. Their speed is now zero so in this frame the magnetic force is



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