Advertisement

Generalization of Noether’s Theorem in Modern Form to Non-variational Partial Differential Equations

  • Stephen C. AncoEmail author
Chapter
Part of the Fields Institute Communications book series (FIC, volume 79)

Abstract

A general method using multipliers for finding the conserved integrals admitted by any given partial differential equation (PDE) or system of partial differential equations is reviewed and further developed in several ways. Multipliers are expressions whose (summed) product with a PDE (system) yields a local divergence identity which has the physical meaning of a continuity equation involving a conserved density and a spatial flux for solutions of the PDE (system). On spatial domains, the integral form of a continuity equation yields a conserved integral. When a PDE (system) is variational, multipliers are known to correspond to infinitesimal symmetries of the variational principle, and the local divergence identity relating a multiplier to a conserved integral is the same as the variational identity used in Noether’s theorem for connecting conserved integrals to invariance of a variational principle. From this viewpoint, the general multiplier method is shown to constitute a modern form of Noether’s theorem in which the variational principle is not directly used. When a PDE (system) is non-variational, multipliers are shown to be an adjoint counterpart to infinitesimal symmetries, and the local divergence identity that relates a multiplier to a conserved integral is shown to be an adjoint generalization of the variational identity that underlies Noether’s theorem. Two main results are established for a general class of PDE systems having a solved-form for leading derivatives, which encompasses all typical PDE systems of physical interest. First, all non-trivial conserved integrals are shown to arise from non-trivial multipliers in a one-to-one manner, taking into account certain equivalence freedoms. Second, a simple scaling formula based on dimensional analysis is derived to obtain the conserved density and the spatial flux in any conserved integral, just using the corresponding multiplier and the given PDE (system). Also, a general class of multipliers that captures physically important conserved integrals such as mass, momentum, energy, angular momentum is identified. The derivations use a few basic tools from variational calculus, for which a concrete self-contained formulation is provided.

Notes

Acknowledgements

S.C. Anco is supported by an NSERC research grant. The referees are thanked for valuable comments which have improved this work.

References

  1. 1.
    P. Olver, Applications of Lie Groups to Differential Equations (Springer-Verlag, New York, 1986).CrossRefzbMATHGoogle Scholar
  2. 2.
    G. Bluman and S.C. Anco, Symmetry and Integration Methods for Differential Equations, Springer Applied Mathematics Series 154 (Springer-Verlag, New York, 2002).Google Scholar
  3. 3.
    G. Bluman, A. Cheviakov, S.C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Springer Applied Mathematics Series 168, (Springer, New York, 2010).CrossRefzbMATHGoogle Scholar
  4. 4.
    L. Martinez Alonso, Lett. Math. Phys. 3, 419–424 (1979).MathSciNetCrossRefGoogle Scholar
  5. 5.
    Y. Kosmann-Schwarzbach, The Noether theorems. Invariance and conservation laws in the twentieth century, Sources and Studies in the History of Mathematics and Physical Sciences (translated, revised, and augmented by B.E. Schwarzbach), (Springer, New York, 2011).Google Scholar
  6. 6.
    S.C. Anco and G. Bluman, Phys. Rev. Lett. 78, 2869–2873 (1997).MathSciNetCrossRefGoogle Scholar
  7. 7.
    S.C. Anco and G. Bluman, Euro. J. Appl. Math. 13, 545–566 (2002).Google Scholar
  8. 8.
    S.C. Anco and G. Bluman, Euro. J. Appl. Math. 13, 567–585 (2002).Google Scholar
  9. 9.
    S.C. Anco, J. Phys. A: Math. and Gen. 36, 8623–8638 (2003).CrossRefGoogle Scholar
  10. 10.
    T. Wolf, Euro. J. Appl. Math. 13, 129–152 (2002).CrossRefGoogle Scholar
  11. 11.
    C. Morawetz, Bulletin Amer. Math. Soc. 37(2), 141–154 (2000).MathSciNetCrossRefGoogle Scholar
  12. 12.
    A.H. Kara and F.M. Mahomed, Nonlin. Dyn. 45, 367–383 (2006).CrossRefGoogle Scholar
  13. 13.
    N.H. Ibragimov, J. Math. Anal. Appl. 333, 311–328 (2007).MathSciNetCrossRefGoogle Scholar
  14. 14.
    N.H. Ibragimov, J. Phys. A: Math. and Theor. 44, 432002 (2011).CrossRefGoogle Scholar
  15. 15.
    D. Poole and W. Hereman, J. Symbolic Computation 46, 1355–1377 (2011).MathSciNetCrossRefGoogle Scholar
  16. 16.
    S.C. Anco and J. Pohjanpelto, Classification of local conservation laws of Maxwell’s equations, Acta. Appl. Math. 69, 285–327 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    I.S. Krasil’shchik and A.M. Vinogradov (eds.), Symmetries and Conseervation Laws for Differential Equations of Mathematical Physics, Translations of Mathematical Monographs 182, Amer. Math. Soc.:Providence, 1999.Google Scholar
  18. 18.
    A. Verbotevsky, in: Secondary Calculus and Cohomological Physicss, 211–232, Contemporary Mathematics 219, Amer. Math. Soc.:Providence, 1997.Google Scholar
  19. 19.
    G. Barnich, F. Brandt, and M. Henneaux, Commun. Math. Phys. 174 (1994), 57–91.CrossRefGoogle Scholar
  20. 20.
    L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics 19, (Amer. Math. Soc., Providence, 1998).Google Scholar
  21. 21.
    A. Degaspersis, A.N.W. Hone, D.D. Holm, in: Nonlinear Physics: Theory and Experiment II, 37–43, (eds. M.J. Ablowitz, M. Boiti, F. Pempinelli, and B. Prinari) World Scientific (2003).Google Scholar
  22. 22.
    L.V. Ovsiannikov, Group Analysis of Differential Equations (Academic Press, New York, 1982).zbMATHGoogle Scholar
  23. 23.
    R.L. Anderson and N.H. Ibragimov, Lie-Bäcklund Transformations in Applications (SIAM Studies in Applied Mathematics, 1979).CrossRefGoogle Scholar
  24. 24.
    N.H. Ibragimov, Transformation groups applied to mathematical physics (Reidel, Dordrecht, 1985).CrossRefzbMATHGoogle Scholar
  25. 25.
    N.H. Ibragimov (ed.), CRC Handbook of Lie Group Analysis of Differential Equations Volumes I,II,III (CRC Press, 1994–1996).Google Scholar
  26. 26.
    J. Nestruev, Smooth manifolds and observables, (Springer, New York, 2002).zbMATHGoogle Scholar
  27. 27.
    S.C. Anco and A. Kara, Euro. J. Appl. Math. (in press) (2017).Google Scholar
  28. 28.
    S.C. Anco, Int. J. Mod. Phys. B 30, 164004 (2016).Google Scholar
  29. 29.
    S.C. Anco, Symmetry 9(3), 33 (2017).CrossRefGoogle Scholar
  30. 30.
    R.S. Khamitara, Teoret, Mat. Fiz. 52(2), 244–251 (1982); English translations Theoret. and Math. Phys. 52(2), 777–781, (1982).Google Scholar
  31. 31.
    N.H. Ibragimov, A.H. Kara, F.M. Mahomed, Non. Dyn. 15(2), 115–136 (1998).CrossRefGoogle Scholar
  32. 32.
    G. Bluman, Tempurchaalu, S.C. Anco, J. Math. Anal. Appl. 322, 233–250 (2006).Google Scholar
  33. 33.
    S.C. Anco and W. Hereman, in preparation.Google Scholar
  34. 34.
    S.C. Anco, in preparation.Google Scholar

Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsBrock UniversitySt. CatharinesCanada

Personalised recommendations