In field theory, if you are given an action $S$, the standard way to find the stress energy tensor is $$ T^{\mu\nu}= -\frac{2}{\sqrt g}\frac{\delta S}{\delta g^{\mu\nu}}.$$ But there are many examples of classical equations of motion that cannot be derived from an action (take first order hydrodynamics as an example). My question is: If you only know the equations of motion of a system (and no corresponding action exists), how do you find the stress energy tensor?


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