Classical field theory of the Von Mises equation for irrotational polytropic inviscid fluids

The Von Mises second-order quasi-linear partial differential equation describesthe dynamics of an irrotational, compressible and barotropic classical perfectfluid through a scalar function only, i.e. the velocity potential. It is shownhere how to derive it in the case of a polytropic equation of state startingfrom a least action principle. The Lagrangian density is found to coincide withpressure.Once re-expressed in terms of the velocity potential, the action integralpresents some similaritieswith other classical and quantum field theories. Aidedby the Legendre transformation tool, we show that the nonlinear equation iscompletely integrable in the case of a non-steady parallel flow dynamics. Anumerical solution of the equation in a critical shock wave forming scenarioallows one to analyze then such a particular dynamics by using the so-calledanalogue gravity formalism which impressively links ordinary perfect fluids tocurved spacetimes. Finally, implications for fluid dynamics and field theoriesare discussed.