On Noether’s Theorem for the Euler–Poincaré Equation on the Diffeomorphism Group with Advected Quantities

• Published in: Foundations of computational mathematics Vol. 13; no. 4; pp. 457 - 477
• Main Authors: Cotter, C. J., Holm, D. D.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.08.2013; Springer; Springer Nature B.V
• Subjects: Applications of Mathematics; Computational mathematics; Computer Science; Conservation laws; Density; Differential equations; Economics; Eulers equations; Evolution; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Symmetry; Theorems; Theorems (Mathematics); Vector space; Vector spaces; Vectors (mathematics); United Kingdom; Conservation laws; Hamiltonian structures; Symmetries; 37K05; Variational principles
• DOI: 10.1007/s10208-012-9126-8
• ISSN: 1615-3375

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 Sect. 4 of Holm et al. (Adv. Math. 137:1–81, 1998 ).