Field Dynamics in a Unified Differential Forms Framework: From Field Strength Tensor to Compressible Flow, Navier-Stokes Equations, and Vorticity Dynamics

At a point p in a field, the Lagrangian density can be expressed as the interior product of the tangent velocity vector \(\overrightarrow{v} \in T_{p}M\) and its corresponding metric dual conjugate momentum 1-form \(S \in T_{p}^{*}M\), i.e., \(\mathcal{L =}S\left( \overrightarrow{v} \right)\). Taking the exterior derivative of this 1-form yields a differential 2-form \(\omega = dS\), whose components constitute the field strength tensor—an antisymmetric (0,2)-tensor. Contracting this 2-form with the tangent velocity vector gives the dynamic equation of the flow (a 1-form): \(\iota_{\overrightarrow{v}}(\omega) = 0\). This formulation is entirely general and does not rely on prior assumptions. In reality, all fields exhibit some degree of compressibility. When this method is applied to a compressible field, it yields the dynamic equations for compressible flow. A singularity arises when the flow velocity is equal to the local wave propagation speed. In the case that the flow velocity is much less than the wave speed, or the wave speed approaches infinity, as an approximation, the dynamic equation degenerates to an incompressible flow. Further, by neglecting local spinning motion and applying Stokes's hypothesis, the equation reduces to the classical Navier-Stokes equations. The second exterior derivative \(d^{2}S = 0\) yields a homogeneous differential 3-form. The coefficients of this 3-form correspond to the dynamic equations governing the vorticity field, providing for the absence of sources, sinks, or singularities at the point under consideration.