Why should the potential of a non-relativistic isolated system be velocity independent?

The lagrangian function of an non-relativistic isolated system of point masses is $$L=\sum_i\frac{m_i}{2}\dot{\vec r}_i^2-V,$$ where the potential function $V$ represents all interactions.

If we assume Newton's principle of determinancy, then by knowing the initial positions and velocities we know any future state of the system. This implies that the potential may be a function of positions, velocities and time, $V=V(\vec r_i,\dot{\vec r}_i,t)$.

Let us now assume Galilean invariance which means the system is invariant under time translations, spatial translations, rotations and change of state of uniform motion.

• Time translation invariance, $t\rightarrow t+s$, implies that the potential does not depend explicitly on time. Hence $V=V(\vec r_i,\dot{\vec r}_i)$.
• Spatial translation invariance, $\vec r_i\rightarrow \vec r_i+\vec a$, implies that the position dependence of the potential is only on the relative vectors, $V=V(\vec r_i-\vec r_j,\dot{\vec r}_i)$.
• If the system is invariant under change of state of uniform motion, $\vec r_i\rightarrow \vec r_i+\vec vt$, then also the velocity dependence must be through relative velocities, $V=V(\vec r_i-\vec r_j,\dot{\vec r}_i-\dot{\vec r}_j)$.
• Invariance under rotations, $\vec r\rightarrow R\vec r$, $R\in SO(3)$, leads to the fact that what is important is length and not orientation, so $V=V(|\vec r_i-\vec r_j|,|\dot{\vec r}_i-\dot{\vec r}_j|)$.

The question is: Should not the potential function of such system be just $V(|\vec r_i-\vec r_j|)$ ? How to eliminate the dependence on $|\dot{\vec r}_i-\dot{\vec r}_j|$ ?

• For more than one particle, rotation invariance does not imply the absolute-value expression you've written down on the last bullet point. – Emilio Pisanty Apr 12 '17 at 6:38
• @EmilioPisanty Would you please elaborate a little bit on that? – Diracology Apr 12 '17 at 11:53
• Actually, even for two particles it's not true: a rotationally invariant potential $V(\vec r_1,\vec r_2,\dot{\vec r}_1,\dot{\vec r}_2)$ can depend on the angle between the relative separation $\vec r_1-\vec r_2$ and the relative velocity $\dot{\vec r}_1-\dot{\vec r}_2$. – Emilio Pisanty Apr 12 '17 at 12:57
• We usually assume the potential depends only on distances but that's just to simplify things. For example, the magnetic force depends on the angle between velocity and displacement. – Javier Apr 12 '17 at 13:29
• @Javier But electromagnetic interaction is relativistic. I might be wrong but I presume there are constraints for non-relativistic system that put the potential as a function of only $|\vec r_i-\vec r_j|$. – Diracology Apr 12 '17 at 13:32