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I know quite a lot of Variational principles (VP) yielding systems of classical mechanics. By a VP, I mean something like $$\delta{\cal L}[U]=0$$ where ${\cal L}$ is a functional and the field belongs to some suitable space and its variations as well. For instance, in $n$-dimensional gas dynamics, the Euler system (conservation of mass, momentum and energy) is associated with the Lagrangian $${\cal L}(\alpha,s)=\int\int\left(\frac{|m|^2}{2\rho}+f(\rho,s)\right)dxdt,$$ where $\alpha$ is a closed $n$-form whose coordinates are $(\rho,m)$ (the mass density and linear momentum) while $s$ is a function, that is a $0$-form (the entropy). The VP writes $$\left.\frac{d}{d\epsilon}\right|_{\epsilon=0}{\cal L}[\phi_\epsilon^*(\alpha,s)]=0$$ for every flow of smooth, compactly supported vector field. Notice that the conservation of mass is encoded in the closedness of $\alpha$ : $\partial_t\rho+{\rm div}_xm=0$. See D.S. Sur le principe variationnel des équations de la mécanique des fluides parfaits. M2AN - Modél. Math. et Anal. Num., 27 (1993), pp 739-758.

My question is whether such a VP is available for the relativistic version of the Euler system, where now we do not distinguish between conservation of mass and energy. This system writes \begin{eqnarray*} \partial_t\left(\frac{\rho c^2+p}{c^2-|v|^2}\,-\,\frac p{c^2}\right)+{\rm div}_x\left(\frac{\rho c^2+p}{c^2-|v|^2}\,v\right) & = & 0, \\ \partial_t\left(\frac{\rho c^2+p}{c^2-|v|^2}\,v\right)+{\rm Div}_x\left(\frac{\rho c^2+p}{c^2-|v|^2}\,v\otimes v\right)+\nabla_x p & = & 0. \end{eqnarray*}

Feel free to assume $p=a^2\rho$ for some positive constant $a$.

Added after Igor's answer. I am particularly interested in formulations that avoid Lagrange multipliers.

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For now I'll just mention that there's a small literature on variational principles for perfect fluids in relativity, though I'm not an expert on it. Here is a reference that discusses some approaches, with references to previous works. It is in the context of general relativity, but it is sufficient to set the metric $g_{\mu\nu}$ to the Minkowski space metric to recover the special relativistic version. Unfortunately, I don't know of a more self-contained source that treats just the special-relativistic version.

Brown, J. David, Action functionals for relativistic perfect fluids, Classical Quantum Gravity 10, No. 8, 1579-1606 (1993). ZBL0788.76101.

Perhaps when I have time, I could add below an explicit version for of one of the action functionals.

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  • $\begingroup$ Interesting ! Would you have a reference which does not make use of agrange multipliers ? $\endgroup$ Aug 21, 2021 at 7:40
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There is by now also quite some literature on action principles for relativistic dissipative fluids. A random reference is this paper. The short answer to your question is that the action for Euler fluids is the fluid pressure.

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