  This paper presents a novel Hamiltonian formulation of the isotropic
Navier-Stokes problem based on a minimum-action principle derived from the
principle of least squares. This formulation uses the velocities
$u_{i}(x_{j},t)$ and pressure $p(x_{j},t)$ as the field quantities to be
varied, along with canonically conjugate momenta deduced from the analysis.
From these, a conserved Hamiltonian functional $H^{*}$ satisfying Hamilton's
canonical equations is constructed, and the associated Hamilton-Jacobi equation
is formulated for both compressible and incompressible flows. This
Hamilton-Jacobi equation reduces the problem of finding four separate field
quantities ($u_{i}$,$p$) to that of finding a single scalar functional in those
fields--Hamilton's principal functional $\text{S}^{*}[t,u_{i},p]$. Moreover,
the transformation theory of Hamilton and Jacobi now provides a prescribed
recipe for solving the Navier-Stokes problem: Find $\text{S}^{*}$. If an
analytical expression for $\text{S}^{*}$ can be obtained, it will lead via
canonical transformation to a new set of fields which are simply equal to their
initial values, giving analytical expressions for the original velocity and
pressure fields. Failing that, if one can only show that a complete solution to
this Hamilton-Jacobi equation does or does not exist, that will also resolve
the question of existence of solutions. The method employed here is not
specific to the Navier-Stokes problem or even to classical mechanics, and can
be applied to any traditionally non-Hamiltonian problem.
