Advertisement

Noether’s Theorems and the Direct Method

  • Gary Webb
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 946)

Abstract

In this chapter we give a general discussion of Noether’s theorems and the Calculus of Variations, for systems of differential equations governed by a variational principle. Noether’s theorems are discussed by Gelfand and Fomin (1963), Ibragimov (1985), Marsden and Ratiu (1994), Holm (2008a,b), Bluman and Kumei (1989)), Olver (1993), Anco and Bluman (1996, 1997, 2002a,b), and Bluman et al. (2010) and others. We use the analysis of Bluman and Kumei (1989) and Ibragimov (1985) as summarized by Webb et al. (2005b). Hydon and Mansfield (2011) give a clear presentation of Noether’s second theorem. The main aim is to briefly present Noether’s theorems and methods to derive conservation laws. Noether’s theorems link the symmetries of the action and conservation laws, for systems of differential equations governed by a variational principle.

References

  1. Anco, S.C., Bluman, G.: Derivation of Conservation Laws from Nonlocal Symmetries of Differential Equations. J. Math. Phys. 37(5), 2361–2375 (1996)Google Scholar
  2. Anco, S.C., Bluman, G.: Direct Construction of Conservation Laws from Field Equations. Phys. Rev. Lett. 78(15), 2869–2873 (1997)Google Scholar
  3. Anco, S.C., Bluman, G.W.: Direct Construction Method for Conservation Laws of Partial Differential Equations. Part I: Examples of Conservation Law Classification. Eur. J. Appl. Math. 13, 545–566 (2002a)Google Scholar
  4. Anco, S.C., Bluman, G.W.: Direct Construction Method for Conservation Laws of Partial Differential Equations. Part II: General Treatment. Eur. J. Appl. Math. 13, 567–585 (2002b)Google Scholar
  5. Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Applied Mathematical Sciences 81. Springer, New York (1989)Google Scholar
  6. Bluman, G.W., Cheviakov, A.F., Anco, S.C.: Applications of Symmetry Methods to Partial Differential Equations. Applied Mathematical Sciences Series 168. Springer, New York (2010)Google Scholar
  7. Cheviakov, A.F.: GeM Software Package for Computation of Symmetries and Conservation Laws of Differential Equations. Comput. Phys. Commun. 176, 48–61 (2007)Google Scholar
  8. Cheviakov, A.F.: Conservation Properties and Potential Systems of Vorticity-Type Equations. J. Math. Phys. 55, 033508 (16 pp.) (2014) (0022-2488/2014/55(3)/033508/16)Google Scholar
  9. Cheviakov, A.F., Anco, S.C.: Analytical Properties and Exact Solutions of Static Plasma Equilibrium Systems in Three Dimensions. Phys. Lett. A 372, 1363–1373 (2008)Google Scholar
  10. Gelfand, I.M., Fomin, S.V.: Calculus of Variations, Translated and Edited by Richard A Silverman, Dover edition 2000. Dover, New York (1963)Google Scholar
  11. Hereman, W., Colagrosso, M., Sayers, R., Ringler, A., DeConninck, B., Nivala, M., Hickman, M.: Continuous and Discrete Homotopy Operators and the Computation of Conservation Laws. In: Wang, D., Zheng, Zh. (eds.) Differential Equations with Symbolic Computation. Trends in Mathematics, Chap. 15, pp. 249–285. Birkhäuser, Basel (2006)Google Scholar
  12. Holm, D.D.: Geometric Mechanics, Part I: Dynamics and Symmetry. Imperial College Press, London (2008a). Distributed by World ScientificGoogle Scholar
  13. Holm, D.D.: Geometric Mechanics, Part II: Rotating, Translating and Rolling. Imperial College Press, London (2008b). Distributed by World ScientificGoogle Scholar
  14. Hydon, P.E., Mansfield, E.L.: Extensions of Noether’s Second Theorem: From Continuous to Discrete Systems. Proc. R. Soc. A 467, 3206–3221 (2011). https://doi.org/https://doi.org/10.1098/rspa.2011.0158Google Scholar
  15. Ibragimov, N.H.: Transformation Groups Applied to Mathematical Physics. Reidel, Dordrecht (1985)Google Scholar
  16. Ibragimov, N.H.: In: Ibragimov, N.H. (ed.) CRC Handbook of Lie Group Analysis of Differential Equations 1. CRC, Boca Raton (1994)Google Scholar
  17. Kara, A.H., Mahomed, F.M.: Relationship Between Symmetries and Conservation Laws. Int. J. Theor. Phys. 39(1), 23–40 (2000)Google Scholar
  18. Kara, A.H., Mahomed, F.M.: A Basis of Conservation Laws for Partial Differential Equations. J. Nonlin. Math. Phys. 9, Suppl. 2, 60–72 (2002)Google Scholar
  19. Kelbin, O., Cheviakov, A.F., Oberlack, M.: New Conservation Laws of Helically Symmetric, Plane and Rotationally Symmetric Viscous and Inviscid Flows. J. Fluid Mech. 721, 340–366 (2013)Google Scholar
  20. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. Springer, New York (1994)Google Scholar
  21. Morse, P.M., Feshbach, H.: Methods of Theoretical Physics, Part I. McGraw Hill, New York (1953)Google Scholar
  22. Noether, E.: Invariante Variations Probleme. Nachr. König. Gessell. Wissen. Göttingen Mathphys. Kl. 2, 235–257 (1918). See Transp. Theory Stat. Phys. 1, 186–207 (1971). For an English translation, posted at physics/0503066Google Scholar
  23. Olver, P.J.: Applications of Lie Groups to Differential Equations, 2nd edn. Springer, New York (1993)Google Scholar
  24. Padhye, N.S.: Topics in lagrangian and hamiltonian fluid dynamics: relabeling symmetry and ion acoustic wave stability. Ph.D. Dissertation, University of Texas at Austin (1998)Google Scholar
  25. Padhye, N.S., Morrison, P.J.: Fluid Relabeling Symmetry. Phys. Lett. A 219, 287–292 (1996a)Google Scholar
  26. Padhye, N.S., Morrison, P.J.: Relabeling Symmetries in Hydrodynamics and Magnetohydrodynamics. Plasma Phys. Rep. 22, 869–877 (1996b)ADSGoogle Scholar
  27. Pshenitsin, D.: Conservation laws of magnetohydrodynamics and their symmetry transformation properties. Ph.D. Thesis, Department of Physics, Brock University, Saint Catharines (2016). Available at https://dr.library.ca.handle/10464/9801
  28. Webb, G.M., Mace, R.L.: Potential Vorticity in Magnetohydrodynamics. J. Plasma Phys. 81, p. 18, 905810115 (2015). https://doi.org/10.1017/S0022377814000658. Preprint: http://arxiv/org/abs/1403.3133Google Scholar
  29. Webb, G.M., Zank, G.P., Kaghashvili, E.Kh., Ratkiewicz, R.E.: Magnetohydrodynamic Waves in Non-uniform Flows II: Stress Energy Tensors, Conservation Laws and Lie Symmetries. J. Plasma Phys. 71, 811–857 (2005b). https://doi.org/10.1017/S00223778050003740 Google Scholar
  30. Webb, G.M., Hu, Q., Dasgupta, B., Zank, G.P.: Homotopy Formulas for the Magnetic Vector Potential and Magnetic Helicity: The Parker Spiral Interplanetary Magnetic Field and Magnetic Flux Ropes. J. Geophys. Res. (Space Phys.) 115, A10112 (2010a). https://doi.org/10.1029/2010JA015513. Corrections: J. Geophys. Res. 116, A11102 (2011). https://doi.org/10.1029/2011JA017286

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Gary Webb
    • 1
  1. 1.CSPARThe University of Alabama in HuntsvilleHuntsvilleUSA

Personalised recommendations