  The Navier-Stokes Hamiltonian is derived from first principles. Its Hamilton
equations are shown to be equivalent to the continuity, Navier-Stokes, and
energy conservation equations of a compressible viscous fluid. The derivations
of the Euler and Navier-Stokes Hamiltonians are compared, with the former
having identical dynamics to the Euler equation with the viscosity terms of the
Navier-Stokes equation dropped from the beginning. The two Hamiltonians have
the same number of degrees of freedom in three spatial and one temporal
dimension: six independent scalar potentials, but their dynamical fields are
necessarily different due to a theory with dissipation not mapping smoothly
onto one without. The dynamical coordinate field of a dissipative fluid is a
vector field that stores the initial position of all of its fluid particles.
Thus these appear to be natural coordinates for studying arbitrary separations
of fluid particles over time. The classical similarity renormalization group is
introduced and the first steps are carried out deriving a flow equation for the
Navier-Stokes Hamiltonian. Finally, the symmetries of a nonrelativistic viscous
fluid are discussed through its galilean algebra with dissipative canonical
Poisson brackets.
