Foundations of Computational Mathematics

, Volume 13, Issue 4, pp 457–477 | Cite as

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

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

Keywords

Hamiltonian structures Symmetries Variational principles Conservation laws 

Mathematics Subject Classification

37K05 
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Notes

Acknowledgements

This paper was inspired by remarks made to C.J.C. by Oliver Bühler about Kelvin’s theorem and the paper [15]. The authors are also grateful to Y. Kosmann-Schwartzbach and E.L. Mansfield for comments, encouragement and advice while we were writing of this paper, and the useful suggestions and corrections from the two anonymous reviewers. The work by D.D.H. was partially supported by an Advanced Grant from the European Research Council.

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Copyright information

© SFoCM 2012

Authors and Affiliations

  1. 1.Aeronautics DepartmentImperial College LondonLondonUK
  2. 2.Mathematics DepartmentImperial College LondonLondonUK

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