# Question on force invariance under the Galilean Transformations (GT)

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 ?

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
$$F=\frac{\mu_0}{4\pi}\frac{q^2v^2}{d^2}.$$
$$F’=0.$$