How to see linearity of an interaction if it's lagrangian density is known?

The Lagrangian of electrodynamics is $$-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+A_\mu J^\mu$$ we know that electrodynamics is linear in special relativity but when we go to general relativity it becomes non-linear.

Another example will be linearised Einstein field equation. It's $$\mathcal{L}=\frac{1}{2}[(\partial_\mu h^{\mu \nu})(\partial_\nu h)-(\partial_\mu h^{\rho \sigma})(\partial_{\rho}h^\mu_\sigma)+\frac{1}{2}\eta ^{\mu \nu}(\partial_\mu h^{\rho \sigma})(\partial_\nu h_{\rho \sigma})-\frac{1}{2}\eta ^{\mu \nu}(\partial_\mu h)(\partial_\nu h)]$$.

So my question is how to see whether a theory is linear (superposition holds) or not by looking at its Lagrangian density?

• classical equation of motion? – FangXie Apr 11 at 18:35

To get the equations of motion out of a Lagrange density you need to calculate the Euler-Lagrange equation $$0 = \left(\frac{\partial}{\partial \phi}-\partial_\mu \frac{\partial}{\partial \partial_\mu \phi}\right) \mathcal L.$$