Lagrangian dynamics of the Navier-Stokes equation

TL;DR

This paper introduces a novel gauge-invariant Lagrangian framework for fluid dynamics that reconstructs the Navier-Stokes equation via the Euler-Lagrange formalism, offering an alternative to traditional solution methods.

Contribution

It develops a gauge-invariant Lagrangian approach to fluid dynamics, connecting the Navier-Stokes equation with field theory concepts, which is a new perspective in the field.

Findings

Reconstruction of Navier-Stokes equation from a gauge-invariant Lagrangian.

Application of the framework to fluid interactions in a solitonic medium.

Establishment of a field-theoretic approach to classical fluid dynamics.

Abstract

Most researches on fluid dynamics are mostly dedicated to obtain the solutions of Navier-Stokes equation which governs fluid flow with particular boundary conditions and approximations. We propose an alternative approach to deal with fluid dynamics using the lagrangian. We attempt to develop a gauge invariant lagrangian which reconstructs the Navier-Stokes equation through the Euler-Lagrange equation. The lagrangian consists of gauge boson field $\A_\mu$ with appropriate content describing the fluid dynamics, i.e. $\A_\mu = (\Phi, -\uv)$. An example of applying the lagrangian to the interaction of fluid in a solitonic medium is also given.