Clebsch Potentials in the Variational Principle for a Perfect Fluid

TL;DR

This paper explains the role of additional fields in variational formulations of perfect fluid dynamics, clarifies their necessity for rotational flows, and derives a Hamiltonian framework using control theory.

Contribution

It provides a simple explanation for the additional fields needed in the Eulerian variational principle and develops a Hamiltonian formulation treating velocity as input.

Findings

Additional fields fix pathline endpoints in variational calculus.

The explanation applies to rotational isentropic flows.

A canonical Hamiltonian formulation is derived using control theory.

Abstract

Equations for a perfect fluid can be obtained by means of the variational principle both in the Lagrangian description and in the Eulerian one. It is known that we need additional fields somehow to describe a rotational isentropic flow in the latter description. We give a simple explanation for these fields; they are introduced to fix both ends of a pathline in the variational calculus. This restriction is imposed in the former description, and should be imposed in the latter description. It is also shown that we can derive a canonical Hamiltonian formulation for a perfect fluid by regarding the velocity field as the input in the framework of control theory.