I'm following a course of Lagrangian and Hamiltonian mechanics, but I'm getting somewhat confused. Could someone explain the difference between $L$ and $\mathcal{L}$?
I'm calling both "the Lagrangian of the system", but they seem te be different?
Also I'm somewhat confused in how to find these functions. The big advantage seems to be that, using the Euler-Lagrange equations you can easily describe the motion of the system. However to find the Lagrangian $L=T-V$ one needs to know the kinetic and potential energy. Which depend on $\dot x, \dot y, \dot z, \ldots$ Seems to me that to find $T$ you already need to know the motion of the system?
I guess I'm starting to mix everything together, could someone clarify?