Is there any physical scalar potential where $U$ is also depends on $\dot q_i$s?

Question

In the book of Classical Mechanics by Goldstein, at page 21, while deriving the Lagrange's equation, when the external forces are derivable from a scalar potential $U$, the author implicitly assumes that $U = U(q_i, t)$.

However, is there any physical scalar potential where $U$ is also depends on $\dot q_i$s ?

Answer 1

Yes, e.g. the velocity-dependent potentials for the Lorentz force and the Coriolis force. See also e.g. this Phys.SE post.