Question

can lagrangian be applied to the cartesian coordinate system? if yes then does the differential of T w.r.t Q(dot) vanishes?

Answer 1

yes it can be applied, \( \dot q \) vanishes if your Lagrangian is a linear function of \( \dot q \) however usually your Lagrangian is a quadratic function of \( \dot q^2\) and usually does not vanish.