# Legendre transform and coordinate system independence

I'm self-learning analytical mechanics.

Consider a classical mechanical system. Even if it's clear to me that via (the usual) Legendre transform we can get a unique Hamiltonian function from a Lagrangian (at least for well suited cases), what does happen if we change the coordinate system? AFAIK the relation has to be independent from coordinate change, but how to prove it?

• – Qmechanic Jul 24 at 14:34
• Thanks @Qmechanic. Are you in particular recommending the 6th point in your answer? – Lo Scrondo Jul 24 at 14:42
• $\uparrow$ Yes. – Qmechanic Jul 24 at 14:44