Hamiltonian and Lagrange density

[kaksmet]

Main Question or Discussion Point

Hello everyone!
I'm trying to find the relation between the lagrangian density and the hamiltonian, does anyone know how they are related? I also need a reference where I can find the relation.

Thanks!

 

[lbrits]

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

 

[kaksmet]

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

Thanks alot! Just one small thing more
If the Hamiltonian is $$\int\!d^3x\, \mathcal{H}$$, then what is $$\mathcal{H}$$? And is $$\mathcal{L}$$ the lagranian or the lagranian density?

 

[lbrits]

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 $$\mathcal{H}$$ is the Hamiltonian, and the integral over $$\mathcal{L}$$ is called the Lagrangian.

 

[kaksmet]

thanks again