LAGRANGIAN AND HAMILTONIAN FORMULATIONS OF COMPRESSIBLE HYDRODYNAMICS

A new formulation of compressible hydrodynamics is presented based on the Lagrange and Hamilton densities script L sign(r,t) = script L sign(∂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 nm v, and energy density 3P/2 of compressible fluids are obtained from a variational principle as functional derivatives of the Lagrange and Hamilton functions L = ∫∫∫script L sign(r,t)d3r and H = ∫∫∫ℋ(r,t)d3r of the gaseous system of volume Ω = ∫∫∫d3r. © 1979 American Institute of Physics.