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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?

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  • $\begingroup$ classical equation of motion? $\endgroup$ – FangXie Apr 11 at 18:35
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To get the equations of motion out of a Lagrange density you need to calculate the Euler-Lagrange equation $$ 0 = \left(\frac{\partial}{\partial \phi}-\partial_\mu \frac{\partial}{\partial \partial_\mu \phi}\right) \mathcal L. $$

In order to arrive at homogeneous linear field equations, the Lagrangian hast to contain exactly second powers of fields and their derivatives, that is kinetic terms and mass terms. Everything else (except for trivial constants) will necessarily lead to non-linear field equations.

If you allow for an inhomogeneity, source terms linear in the field are also possible.

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