We assume that the forces depend on coordinates and velocities both and are such that they can be derived from a generalized potential \( U \) satisfying

$$ \sum_α \vec F_α^{(e)} \frac {\partial \vec r_α} {\partial q_j} = - \frac {\partial U} {\partial q_j} + \frac d {dt} \frac {\partial U} {\partial \dot q_j} \hspace{7em} (33)$$

where \( U \) is a function of \( q, \dot q, t \)
Then again (29) can be written in the Lagrangian form.

$$ L = \frac d {dt} \left( \frac {\partial L} {\partial \dot q_j} \right) - \frac {\partial L} {\partial q_j} = 0 \hspace{7em} (34) $$

where we again have

$$ L = T − U \hspace{7em} (35) $$

We leave verification of (34) as a simple exercise for the reader. \( L \) is called Lagrangian and is a function of generalized coordinates \( q_j \) , generalized velocities \( \dot q_j \) and \( t \)

$$ L = L (q, \dot q, t) \hspace{7em} (36) $$

We shall see that a description of motion of charges in electric and magnetic field requires use of a velocity dependent generalized potential recall that the magnetic forces are velocity dependent.