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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 ?

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

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  • $\begingroup$ my technical knowledge on EMT is very limited, so I couldn't think that.Thanks for the answer. $\endgroup$ – onurcanbektas Feb 15 at 20:31

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