Navier-Stokes Hamiltonian
Billy D. Jones · arXiv (Cornell University) · 2014
The Navier-Stokes Hamiltonian is derived from first principles. Its Hamilton equations are shown to be equivalent to the Navier-Stokes, continuity, and energy conservation equations of standard fluid mechanics. The derivations of the Navier-Stokes and Euler Hamiltonians are compared, with the latter having identical dynamics to the Euler equation with the viscosity terms dropped from the beginning. The two Hamiltonians have the same number of degrees of freedom in three spatial dimensions: six independent scalar potentials (although in the Navier-Stokes case the potentials are two vector fields), but their dynamical fields are necessarily different due to a theory with dissipation not mapping smoothly onto one without. Mass, momentum, and energy conservation give rise to the standard nonholonomic constraint on entropy which is used to construct the Navier-Stokes Hamiltonian. The Newton, Euler-Lagrange, and Hamilton sets of equations are shown to be equivalent for both of these Euler and Navier-Stokes fluids. The dynamical coordinate field of a dissipative fluid is a vector field that stores the initial position of all the fluid particles. Thus these appear to be natural coordinates for studying arbitrary separations of fluid particles over time. The final section discusses energy conservation of the Navier-Stokes Hamiltonian and derives its Poisson bracket with a general classical dissipative observable to set up later work with the similarity renormalization group.