Calculating the Equations of motion for a scalar field

Question

I am recently trying to get some understanding of Quantum Field Theory, therefore I am reading Quantum Field Theory and the Standard Model by M.D. Schwartz. The author takes for an example the following heuristical lagrangian for the graviton as a simple scalar field theory $$ \begin{align*} \mathcal L = - \frac{1}{2} h \Box h + \frac{1}{3}\lambda h^3 + Jh \text{ where }\Box = \partial_\mu^2 . \end{align*} $$ And I am embarrassingly struggling to calculate the equation of motion of this field by using the Euler-Lagrange-Equation. (The problem might be that I am not quite sure what $J$ is in this case.)

If somebody could sketch the calculation to help me that would be very kind.

Answer 1

The first term will yield the D'Alambertian $\Box$ the second: $λh^{2}$ and the third $J$. So equation of motion will be: $$ \Box h = λh^{2} + J $$

You can use the Euler Lagrange equation or straightforward vary the action. The two last terms are trivial the first term will be:

$$δ(h\Box h) =δh\Box h + h\Box δh $$

The first term of R.H.S is ready: $δh$ is already a multiplying factor. You will now need to perform integration by parts and cancel total divergence terms to make $δh$ a multiplying factor on the second term. Can you go on from here? (Note that your Lagrangian depends upon second derivatives of the dynamical variable $h$, so be careful with Euler-Lagrange equations!)