# Energy of Scalar Field

## Homework Statement

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
$$\frac{\delta{}L}{\delta{}\dot{\phi}(\vec{x})}$$?

## 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?

Matterwave
Gold Member
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}##

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

Matterwave