On Noether’s Theorem for the Euler–Poincaré Equation on the Diffeomorphism Group with Advected Quantities
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
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).
- H.D.I. Abarbanel, D.D. Holm, Nonlinear stability of inviscid flows in three dimensions: incompressible fluids and barotropic fluids, Phys. Fluids 30, 3369–3382 (1987). CrossRef
- V.I. Arnold, Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid, Sov. Math. 6, 773–777 (1965).
- V.I. Arnold, B.A. Khesin, Topological Methods in Hydrodynamics (Springer, Berlin, 1998).
- D. Bak, D. Cangemi, R. Jackiw, Energy-momentum conservation in gravity theories, Phys. Rev. D 49(10), 5173–5181 (1994). CrossRef
- T.B. Benjamin, P.J. Olver, Hamiltonian structure, symmetries and conservation laws for water waves, J. Fluid Mech. 125, 137–185 (1982). CrossRef
- A.J. Brizard, Noether derivation of exact conservation laws for dissipationless reduced-fluid models, Phys. Plasmas 17, 112503 (2010). CrossRef
- C.J. Cotter, D.D. Holm, Continuous and discrete Clebsch variational principles, Found. Comput. Math. 9(2), 221–242 (2009). CrossRef
- C.J. Cotter, D.D. Holm, P.E. Hydon, Multisymplectic formulation of fluid dynamics using the inverse map, Proc. R. Soc. A, 463 (2007).
- R.L. Dewar, Hamilton’s principle for a hydromagnetic fluid with a free boundary, Nucl. Fusion 18, 1541–1553 (1978). CrossRef
- H. Ertel, Ein neuer hydrodynamischer Wirbelsatz, Meteorol. Z. Braunschw. 59, 277–281 (1942).
- F. Gay-Balmaz, T.S. Ratiu, Clebsch optimal control formulation in mechanics, J. Geom. Mech. 3(1), 41–79 (2011). CrossRef
- D.D. Holm, Lyapunov stability of ideal compressible and incompressible fluid equilibria in three dimensions, in Hamiltonian Structure and Lyapunov Stability for Ideal Continuum Dynamics, ed. by D.D. Holm, J.E. Marsden, T.S. Ratiu (University of Montreal Press, Montreal, 1994), pp. 125–208.
- D.D. Holm, Euler–Poincaré dynamics of perfect complex fluids, in Geometry, Mechanics, and Dynamics, ed. by P. Newton, P. Holmes, A. Weinstein, (Springer, New York, 2002), pp. 169–180. CrossRef
- D.D. Holm, J.E. Marsden, Momentum maps and measure valued solutions of the Euler–Poincaré equations for the diffeomorphism group, Prog. Math. 232, 203–235 (2004). nlin.CD/0312048. CrossRef
- D.D. Holm, J.E. Marsden, T.S. Ratiu, The Euler–Poincaré equations and semidirect products with applications to continuum theories, Adv. Math. 137, 1–81 (1998). chao-dyn/9801015. CrossRef
- D.D. Holm, J.E. Marsden, T.S. Ratiu, A. Weinstein, Nonlinear stability of fluid and plasma equilibria, Phys. Rep. 123, 1–116 (1985). CrossRef
- D.D. Holm, J.T. Ratnanather, A. Trouvé, L. Younes, Soliton dynamics in computational anatomy, NeuroImage 23, 170–178 (2004). CrossRef
- P.E. Hydon, E.L. Mansfield, Extensions of Noether’s second theorem: from continuous to discrete systems. arXiv:1103.3267v1.
- Y. Kosmann-Schwarzbach, Les Théorèmes de Noether (Éditions de École Polytechnique, Palaiseau, 2004). English translation, 2011.
- C. Muriel, J.L. Romero, P.J. Olver, Variational C ∞ symmetries and Euler–Lagrange equations, J. Differ. Equ. 222, 164–184 (2006). CrossRef
- E. Noether, Invariante variations probleme, Nachr. König. Gessell. Wissen. Göttingen, Mathphys. Kl., 235–257 (1918). See Transp. Theory Stat. Phys. 1, 186–207 (1971) for an English translation, which is also posted at physics/0503066.
- P.J. Olver, Conservation laws in elasticity, I: general results, Arch. Ration. Mech. Anal. 85, 119–129 (1984).
- P.J. Olver, Conservation laws in elasticity, II: linear homogeneous isotropic elastostatics, Arch. Ration. Mech. Anal. 85, 131–160 (1984). CrossRef
- P.J. Olver, Conservation laws in elasticity, III: planar linear anisotropic elastostatics, Arch. Ration. Mech. Anal. 85, 167–181 (1984).
- P.J. Olver, Noether’s theorems and systems of Cauchy–Kovalevskaya type, in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, vol. 23, ed. by B. Nicholaenko, D.D. Holm, J.M. Hyman (Am. Math. Soc., Providence, 1986), pp. 81–104.
- P.J. Olver, Applications of Lie Groups to Differential Equations (Springer, New York, 1993). CrossRef
- N. Padhye, P.J. Morrison, Fluid element relabeling symmetry, Phys. Lett. A 219, 287–292 (1996). CrossRef
- N. Padhye, P.J. Morrison, Relabeling symmetries in hydrodynamics and magnetohydrodynamics, Plasma Phys. Rep. 22, 869–877 (1996).
- D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J.E. Marsden, M. Desbrun, Structure-preserving discretization of incompressible fluids, Physica D 240(6), 333–458 (2011). CrossRef
- P.L. Similon, Conservation laws for relativistic guiding-center plasma, Phys. Lett. A 112(1), 33–37 (1985). CrossRef
- D.E. Soper, Classical Field Theory (Wiley, New York, 1976).
- On Noether’s Theorem for the Euler–Poincaré Equation on the Diffeomorphism Group with Advected Quantities
Foundations of Computational Mathematics
Volume 13, Issue 4 , pp 457-477
- Cover Date
- Print ISSN
- Online ISSN
- Springer US
- Additional Links
- Hamiltonian structures
- Variational principles
- Conservation laws