What is the physical significance of this generalised potential?

Question

Consider a generalised potential of the form $U=-f\vec{v}\cdot\vec{r}$ where $f$ is a constant. This potential should not contribute any internal forces between particles as \begin{equation} \vec{F}=-\nabla_{\vec{r}} U+\frac{d}{dt}\nabla_{\vec{v}}U=0 \end{equation} Physically then this potential does nothing so why is it even ever considered?

Answer 1

For what it's worth, OP's velocity-dependent/generalized potential $U=-\frac{d(f|\vec{r}|^2/2)}{dt}$ is a total time-derivative, and therefore does not contribute to the Euler-Lagrange (EL) equations.