Euler's fluid equations: Optimal Control vs Optimization

TL;DR

This paper reveals that an image-processing optimization approach and classical fluid dynamics optimal control both lead to Euler's equations for incompressible flow, highlighting the role of gauge freedom and symmetry in fluid mechanics.

Contribution

It demonstrates that different variational principles, one from image processing and one from fluid dynamics, produce the same Euler equations due to gauge and symmetry considerations.

Findings

Optimization in image processing implies Euler's equations for fluid flow.

Different Lagrangian parcel dynamics can produce identical Euler equations.

Gauge freedom and relabeling symmetry explain the equivalence of these formulations.

Abstract

An optimization method used in image-processing (metamorphosis) is found to imply Euler's equations for incompressible flow of an inviscid fluid, without requiring that the Lagrangian particle labels exactly follow the flow lines of the Eulerian velocity vector field. Thus, an optimal control problem and an optimization problem for incompressible ideal fluid flow both yield the \emph {same} Euler fluid equations, although their Lagrangian parcel dynamics are \emph{different}. This is a result of the \emph{gauge freedom} in the definition of the fluid pressure for an incompressible flow, in combination with the symmetry of fluid dynamics under relabeling of their Lagrangian coordinates. Similar ideas are also illustrated for SO(N) rigid body motion.