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

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...

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Bibliographic Details
Published inFoundations of computational mathematics Vol. 13; no. 4; pp. 457 - 477
Main Authors Cotter, C. J., Holm, D. D.
Format Journal Article
LanguageEnglish
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
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Summary: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 ).
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-012-9126-8