I wish to know if there is any way to get the convective derivative of the 4-velocity $$ u^{\mu}\partial_{\mu}u^{\nu} $$

from a lagrangian density $L(u^\mu,\partial_{\nu}u^{\mu})$ in order to describe the equation of motion for a relativistic fluid. This equation, for the forceless case is $$ u^{\mu}\partial_{\mu}u^{\nu} = 0. $$

The Euler-Lagrange equation would be $$ \partial_{\nu}\dfrac{\partial L}{\partial \partial_{\nu}u^{\mu}} = \dfrac{\partial L}{\partial u^{\mu}} $$

So, how could I get the second equation from the third one?

Furthermore, I wanted to know if it would be possible to include an electromagnetic interaction term $$ j_{(mass)}^{\mu}\partial_{\mu}u^{\nu} = F^{\nu}_{\mu}j_{(charge)}^{\mu} $$

but that's a bit secondary



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