Let's say the lagrange function of my system is $L = T(z,\dot z) - m g z$ and I want to determine the equations of motion. Why is $\frac{\partial L}{\partial \dot z} = \frac{\partial T(z, \dot z) }{\partial \dot z}- 0$ ?
I want to know why the second term is 0 even though it is obviously dependent on $\dot z$, since $ \frac{d}{dt} z = \dot z$. Can I just always treat $z$ as if it's an independent variable for these purposes?
