Galilean Transformation

In the first inertial system (gray) a particle has the coordinates $\mathbf{r}'=\mathbf{r}+\mathbf{u} \cdot t$. In the second inertial system (black) the same particle has coordinates $\mathbf{r}$. The system moves with velocity $\mathbf{u}$ with respect to the black system. Both systems are inertial systems, this means Newtons laws are valid in both systems \begin{align} m \cdot \ddot{\mathbf{r}}_i = \sum_j \mathbf{F}_{ij}(\mathbf{r}_i-\mathbf{r}_j) \hspace{1cm} \text{ and } \hspace{1cm} m \cdot \ddot{\mathbf{r}}_i' = \sum_j \mathbf{F}_{ij}(\mathbf{r}_i'-\mathbf{r}_j') \end{align} Since only the distance between particles are relevant for the force, the transformation term cancels \begin{align} \mathbf{r}'=\mathbf{r}+\mathbf{u} \cdot t \, \Rightarrow \, \mathbf{r}_i-\mathbf{r}_j=\mathbf{r}_i'-\mathbf{r}_j' \end{align} From (1) follows that the Lagrangian of an inertial system is independent of the location and direction of $\mathbf{r}$, the time and the direction of $\mathbf{v}$. Thus the Lagrangian is a function of the scalar $\mathbf{v}^2=v^2$ only and is independent of $\mathbf{r}$ \begin{align} \frac{\partial L}{ \partial \mathbf{r}}=0 \end{align} thereafter \begin{align} \frac{d}{dt}\frac{\partial L}{ \partial \mathbf{v}}=0 \end{align} and it must hold that $\frac{\partial L}{ \partial \mathbf{v}}=const.$, which means $v$ must be constant. Hence, the only allowed transformation that satisfy (1) are those between coordinate frames moving with constant velocity $v$ with respect to one another and are called Galilean transformation.