Navier–Stokes dynamics on a differential one-form
Troy Story
The Navier–Stokes equations have been solved by transforming the dynamic Navier–Stokes equation into a differential one-form on an odd-dimensional differentiable manifold and then using the principle that this one-form predicts, by analysis with exterior calculus, a set of characteristic differential equations and vortex vector characteristic of Hamiltonian geometry. By contracting this differential one-form with the vortex vector, the Lagrangian was obtained. This solution 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 showed that it is bounded, and is physically reasonable since the square of the gradient of the principal function is bounded.