Conjugate flow action functionals

TL;DR

This paper introduces a novel method to construct action functionals for non-potential field theories by utilizing a conjugate flow of curvilinear coordinates, enabling derivation from a stationary action principle.

Contribution

It presents a general operator framework for defining conjugate flows that symmetrize field equations, allowing the formulation of action functionals for complex nonlinear PDEs.

Findings

Developed a method to derive action functionals for scalar and vector fields.

Identified transformation groups that leave the action invariant.

Potential to discover new conservation laws in fluid dynamics.

Abstract

We present a new general method to construct an action functional for a non-potential field theory. The key idea relies on representing the governing equations of the theory relative to a diffeomorphic flow of curvilinear coordinates which is assumed to be functionally dependent on the solution field. Such flow, which will be called the conjugate flow of the theory, evolves in space and time similarly to a physical fluid flow of classical mechanics and it can be selected in order to symmetrize the Gateaux derivative of the field equations with respect to suitable local bilinear forms. This is equivalent to requiring that the governing equations of the field theory can be derived from a principle of stationary action on a Lie group manifold. By using a general operator framework, we obtain the determining equations of such manifold and the corresponding conjugate flow action functional. In particular, we study scalar and vector field theories governed by second-order nonlinear partial differential equations. The identification of transformation groups leaving the conjugate flow action functional invariant could lead to the discovery of new conservation laws in fluid dynamics and other disciplines.