Homework Help: Energy of Scalar Field

1. Dec 4, 2014

quantum_smile

1. The problem statement, all variables and given/known data
My question is just about a small mathematical detail, but I'll give some context anyways.
(From Rubakov Sec. 2.2)
An expression for energy is given by
$$E= \int{}d^3x\frac{\delta{}L}{\delta{}\dot{\phi}(\vec{x})}\dot{\phi}(\vec{x}) - L,$$
where L is the Lagrangian,
$$L=\int{}d^3{}x(\frac{1}{2}\dot{\phi}^2-\frac{1}{2}\partial_i\phi\partial_i\phi-\frac{m^2}{2}\phi^2).$$
To derive the expression for energy, Rubakov says that
$$\frac{\delta{}L}{\delta{}\dot{\phi}(\vec{x})}=\dot{\phi}(\vec{x}).$$
What I want to know is, simply, how does he get this expression for
$$\frac{\delta{}L}{\delta{}\dot{\phi}(\vec{x})}$$?
2. Relevant equations

3. The attempt at a solution
If I evaluate the expression, I just get
$$\delta{}L=\int{}d^3x(\dot{\phi}).$$

Where'd the integral go in Rubakov's expression?

2. Dec 4, 2014

Matterwave

I may be wrong, but I believe what is happening is a confusion between the Lagrangian and the Lagrangian density. Look at the expression for the energy, it has an integral in it, so probably the $L$ which appears in there should actually be the Lagrangian density $\mathcal{L}$ defined by $L=\int d^3x \mathcal{L}$

3. Dec 4, 2014

quantum_smile

Ah, that would make a lot of sense (and fix the weird unit problem). Maybe there's a tiny typo in the text.

4. Dec 4, 2014

Matterwave

A lot of field theory texts refer to the Lagrangian density as simply the "Lagrangian", so the language might be confusing. Usually the notation is used so that the Lagrangian density is in a calligraphic font though.