Navier-Stokes dynamics on a differential one-form

After transforming the Navier-Stokes dynamic equation into a characteristic differential one-form on an odd-dimensional differentiable manifold, exterior calculus is used to construct a pair of differential equations and tangent vector(vortex vector) characteristic of Hamiltonian geometry. A solution to the Navier-Stokes dynamic equation is then obtained by solving this pair of equations for the position $x^k $ and the conjugate to the position ${\mathbf{b}}_k $ as functions of time. The solution ${\mathbf{b}}_k $ is shown to be divergence-free by contracting the differential 3-form corresponding to the divergence of the gradient of the velocity with a triple of tangent vectors, implying constraints on two of the tangent vectors for the system. Analysis of the solution ${\mathbf{b}}_k $ shows it is bounded since it remains finite as $\left| {x^k } \right| \to \,\infty $, and is physically reasonable since the square of the gradient of the principal function is bounded. By contracting the characteristic differential one-form with the vortex vector, the Lagrangian is obtained.