On Noether's theorem for the Euler-Poincar\'e equation on the diffeomorphism group with advected quantities

TL;DR

This paper demonstrates how Noether's theorem applies to the Euler-Poincaré equations for ideal fluids with advected quantities, deriving conservation laws directly from symmetries without Lagrangian variables.

Contribution

It provides a Lagrangian-free method to derive conservation laws from symmetries in fluid dynamics with advected quantities, clarifying the role of constraints and their evolution.

Findings

Conservation laws can be derived from particle relabelling symmetries.

Constraints on vector fields preserve advected quantities over time.

Relationship between Noether-derived and other fluid conservation laws is clarified.

Abstract

We show how Noether conservation laws can be obtained from the particle relabelling symmetries in the Euler-Poincar\'e theory of ideal fluids with advected quantities. All calculations can be performed without Lagrangian variables, by using the Eulerian vector fields that generate the symmetries, and we identify the time-evolution equation that these vector fields satisfy. When advected quantities (such as advected scalars or densities) are present, there is an additional constraint that the vector fields must leave the advected quantities invariant. We show that if this constraint is satisfied initially then it will be satisfied for all times. We then show how to solve these constraint equations in various examples to obtain evolution equations from the conservation laws. We also discuss some fluid conservation laws in the Euler-Poincar\'e theory that do not arise from Noether symmetries, and explain the relationship between the conservation laws obtained here, and the Kelvin-Noether theorem given in Section 4 of Holm, Marsden and Ratiu, {\it Adv. in Math.}, 1998.