# Hamiltonian and Lagrange density

1. May 24, 2008

### kaksmet

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!

2. May 24, 2008

### 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
$$\int\!d^3x\, \mathcal{H}$$

3. May 24, 2008

### kaksmet

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?

Last edited: May 24, 2008
4. May 24, 2008

### 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.

5. May 24, 2008

thanks again