# Variational principle for relativistic gas dynamics

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.

## 2 Answers

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.

• Interesting ! Would you have a reference which does not make use of agrange multipliers ? Aug 21, 2021 at 7:40

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.