Hamiltonian, Lagrangian, Dynamics and Singularity of the Compressible Fluid Flow

The wave travels in the compressible fluid by a finite propagation speed. In the center-of-linear-momentum reference frame, the macroscopic velocity is zero, thereby, the macroscopic kinetic energy is also zero, the potential energy density (pressure) and the mass density are equivalent, and the proportional coefficient is the square of the wave speed. The macroscopic dynamic behavior of the compressible fluid flow cannot be depicted in CoM frame but is preferably described in the Lab frame – a pseudo-inertial frame. The system’s velocity is, thus, reference frame dependent, however, wave propagation speed is frame independent. This relative velocity leads the fluid to be compressed and the mass density will increase, and the energy will increase, too, observed in the Lab frame. The increase factor is the Lorentz factor, which depends on the particle’s relative flow velocity (relative to the reference frame) and the wave speed. This system is similar to a variable-mass system. The Hamiltonian, kinetic, and potential energy densities are not only the function of the relative velocity but also the function of wave speed. It is highlighted that for compressible fluids, when the flow velocity is equal to the wave speed, there exists a singularity, where the mass density, kinetic energy, and Hamiltonian increase infinitely great, while the potential energy goes to zero. This behavior is quite different from incompressible flow. According to the definition, the wave propagation speed is infinitely great for incompressible flow, and the potential energy is purely a function of position in flow field, not a function of wave speed. Any change will instantaneously propagate through the whole field without any time lag. The mathematical description cannot depict and thus disregards the wave behavior, just like the Newtonian mechanics. This is also the reason why the particles are instantaneously entangled in quantum mechanics, no matter how far they are in the field. At last, the dynamic equation is given out in Euler coordinates. It shows that the equation is not defined at the transonic point of \(\beta = \frac{v}{c} = 1\) , where there is a singularity point.