A canonical Hamiltonian formulation of the Navier-Stokes problem

TL;DR

This paper introduces a novel Hamiltonian formulation for the Navier-Stokes equations, transforming the problem into solving a Hamilton-Jacobi equation for a scalar functional, which could provide new analytical and existence insights.

Contribution

It develops a Hamiltonian and Hamilton-Jacobi framework for Navier-Stokes, enabling potential analytical solutions and solution existence analysis for fluid dynamics problems.

Findings

Formulated a Hamiltonian functional for Navier-Stokes equations.

Derived a Hamilton-Jacobi equation reducing the problem to a scalar functional.

Proposed a method to analyze solution existence through this framework.

Abstract

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.