Abstract: 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 xk and the conjugate to the position bk as functions of time. The solution bk 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 bk shows it is bounded since it remains finite as | xk |  ∞, 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.