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

Abstract

We show how Noether conservation laws can be obtained from the particle relabelling symmetries in the Euler-Poincaré 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-Poincaré 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.

16 pages, no figures, to appear in Foundations of Computational Mathematics special volume in honor of Peter Olver's 60th birthday

References in corpus (3)

• Soliton Dynamics in Computational Anatomy
• Multisymplectic formulation of fluid dynamics using the inverse map
• Continuous and discrete Clebsch variational principles

Cited by in corpus (8)

• Local and Nonlocal Advected Invariants and Helicities in Magnetohydrodynamics and Gas Dynamics I: Lie Dragging Approach
• Local and Nonlocal Advected Invariants and Helicities in Magnetohydrodynamics and Gas Dynamics II: Noether's Theorems and Casimirs
• Potential Vorticity in Magnetohydrodynamics
• Variational formulations of sound-proof models
• Helicity in Superfluids: existence and the classical limit
• Variational nonlinear WKB in the Eulerian frame
• Euler--Poincaré reduction and the Kelvin--Noether theorem for discrete mechanical systems with advected parameters and additional dynamics
• Helicity in dispersive fluid mechanics