Lagrangian and Hamiltonian formulations of compressible hydrodynamics

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Abstract
A new formulation of compressible hydrodynamics is presented based on the Lagrange and Hamilton densities L (r,t) =L (∂q/∂t,∇∂q) and H (r,t) =H (p,∇⋅q), in which the canonical conjugate variables p and q are given by p=mv and ∂q/∂t=nv [particle mass: m, density field: n (r,t), velocity field: v(r,t)]. Viscous momentum transfer is neglected and conservation of energy is taken into consideration in the polytropic approximation with polytropic coefficient γ〉〈cp/ cV (quasi‐perfect gas). The conservation equations for particle density n, momentum density nmv, and energy density 3P/2 of compressible fluids are obtained from a variational principle as functional derivatives of the Lagrange and Hamilton functions L=F F F L (r,t) d3r and H=F F F H (r,t) d3r of the gaseous system of volume Ω=F F Fd3r.

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