Hamiltonian and Lagrange density

Classical Mechanics by Goldstein covers it in the later chapters, as does the book by Landau. But...
[tex]\mathcal{H} = \pi \dot{\phi} - \mathcal{L}[/tex],
where
[tex]\pi = \frac{\partial\mathcal{L}} { \partial ( \partial_t \phi ) }[/tex]
The Hamiltonian is then
[tex]\int\!d^3x\, \mathcal{H}[/tex]

 

A density is something that you integrate over (like mass density to get mass, charge density to get charge). So the scripted values here are the Hamiltonian and Lagrangian densities, although people often just call them the Hamiltonian and Lagrangian. Usually you can figure out from context.

The integral over [tex]\mathcal{H}[/tex] is the Hamiltonian, and the integral over [tex]\mathcal{L}[/tex] is called the Lagrangian.