Q: How to identify interaction from a Lagrangian In the following Lagrangian of a scalar field $\eta$, one claims that the third and fourth terms are associated to self-interaction triple and quartic coupling vertices,
$$\mathcal{L} = \frac{1}{2} (\partial_\mu \eta)(\partial^\mu \eta) - \lambda v^2 \eta^2 - \lambda v \eta^3 - \frac{1}{4} \lambda \eta^4 + \frac{1}{4} \lambda v^4$$
with $v$ and $\lambda >0$ are constants.
Naively, I would think that terms with two different fields of the form, $\epsilon\eta_1^2\eta_2^2$ ($\epsilon =$ constant) would mean an interaction of two fields through quartic coupling with coupling $\epsilon$, is this correct? If so, is there any direct rule to distinguish all interactions from Lagrangians? Also, what about terms with the derivatives of the fields, e.g. $\epsilon\eta_1(\partial_\mu \eta_2)$?
