When defining $L=T-V$, and using Euler-Lagrange equations ($\partial_x L = \frac{d}{dt} \partial_{\dot{x}} L$), we get back $m \ddot{x} = - \frac{dV}{dx}$ ONLY WHEN ASSUMING that $V(x, \dot{x}) = V(x)$ and $T(x, \dot{x}) = T(\dot{x})$.
However, these assumtions aren't always true (if I'm not mistaken, it may be the case in the electromagnetic potential).
Thus, my question is : why is $L=T-V$ even when those assumptions break?
