Hi I began to study the basics of QED. Now I am studying Lagrangian and Hamiltonian densities of Dirac Equation. I'll call them L density and H density for convenience :) Anyway, the derivation of the H density from L density using Legendre transformation confuses me :( I thought because parameters of them are space-time components, it should be But I found that this is related to the De Doner - Weyl Theory, and the H density used in textbook is where the dot represents time derivative. So, my question is, why we consider Legendre transformation on only time derivative of phi ? Is it just 'defined' to consider energy of the system? Then what does the covariant H density defined in the De Donder - Weyl theory mean?