Confusing Goldstein Statement about Magnitude of the Lagrangian

Question

On page 345 of Goldstein's Classical Mechanics 3rd Ed., he writes:

...the Hamiltonian is dependent both in magnitude and in functional form upon the initial choice of generalized coordinates. For the Lagrangian, we have a specific prescription, $L=T-V$, and a change of generalized coordinates within that prescription may change the functional appearance of $L$ but cannot alter its magnitude.

This statement seems to imply that the Lagrangian is a conserved quantity, which I know is not just incorrect, but wildly incorrect. What is Goldstein trying to say here?

Answer 1

Briefly speaking, Goldstein is trying to convey that the Lagrangian $L(q,\dot{q},t)$ is invariant under passive change of generalized coordinates, cf. e.g. this Phys.SE post; while the Hamiltonian is not always invariant, cf. e.g. this Phys.SE post.