Dear vanhees71, thanks for your comment. I perfectly agree with you that the important difference between the energy and the Lagrangian in this context is that the energy is the first integral and is conserved, whereas the Lagrangian is changing somehow with the motion of the particle.
However...
Hi, I am confused by a point which should be relatively simple. When we consider classical motion of a particle in a central field U(r), we write the total energy E = T + U, where T is the kinetic energy. The kinetic energy contains initially r, r' and φ' (where ' denotes the time derivative)...