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

The Von Mises second-order quasi-linear partial differential equation describes the dynamics of an irrotational, compressible and barotropic classical perfect fluid through a scalar function only, i.e. the velocity potential. It is shown here how to derive it in the case of a polytropic equation of state starting from a least action principle. The Lagrangian density is found to coincide with pressure. Once re-expressed in terms of the velocity potential, the action integral presents some similarities with other classical and quantum field theories. Aided by the Legendre transformation tool, we show that the nonlinear equation is completely integrable in the case of a non-steady parallel flow dynamics. A numerical solution of the equation in a critical shock wave forming scenario allows one to analyze then such a particular dynamics by using the so-called analogue gravity formalism which impressively links ordinary perfect fluids to curved spacetimes. Finally, implications for fluid dynamics and field theories are discussed.