Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 1 (2005), 011, 16 pages      math-ph/0511035      https://doi.org/10.3842/SIGMA.2005.011

Connections Between Symmetries and Conservation Laws

George Bluman
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T1Z2

Received July 29, 2005, in final form October 22, 2005; Published online October 26, 2005

Abstract
This paper presents recent work on connections between symmetries and conservation laws. After reviewing Noether's theorem and its limitations, we present the Direct Construction Method to show how to find directly the conservation laws for any given system of differential equations. This method yields the multipliers for conservation laws as well as an integral formula for corresponding conserved densities. The action of a symmetry (discrete or continuous) on a conservation law yields conservation laws. Conservation laws yield non-locally related systems that, in turn, can yield nonlocal symmetries and in addition be useful for the application of other mathematical methods. From its admitted symmetries or multipliers for conservation laws, one can determine whether or not a given system of differential equations can be linearized by an invertible transformation.

Key words: conservation laws; linearization; nonlocal symmetries; Noether's theorem.

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References

  1. Noether E., Invariante Variationsprobleme, Nachr. König. Gesell. Wissen. Göttingen, Math.-Phys. Kl., 1918, 235-257.
  2. Bessel-Hagen E., Über die Erhaltungssätze der Elektrodynamik, Math. Ann., 1921, V.84, 258-276.
  3. Ibragimov N.H., Transformation groups applied to mathematical physics, Boston, Reidel, 1985.
  4. Olver P.J., Applications of Lie groups to differential equations, New York, Springer, 1986.
  5. Bluman G., Kumei S., Symmetries and differential equations, New York, Springer, 1989.
  6. Anco S., Bluman G., Direct construction of conservation laws from field equations, Phys. Rev. Lett., 1997, V.78, 2869-2873.
  7. Anco S., Bluman G., Integrating factors and first integrals for ordinary differential equations, European J. Appl. Math., 1998, V.9, 245-259.
  8. Anco S., Bluman G., Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classfications, European J. Appl. Math., 2002, V.13, 545-566.
  9. Anco S., Bluman G., Direct construction method for conservation laws of partial differential equations. Part II: General treatment, European J. Appl. Math., 1998, V.9, 245-259.
  10. Krasil'shchik I.S., Vinogradov A.M., Symmetries and conservation laws for differential equations of mathematical physics, Providence, RI, Amer. Math. Soc., 1999.
  11. Wolf T., A comparison of four approaches to the calculation of conservation laws, European J. Appl. Math., 2002, V.13, 129-152.
  12. Bluman G., Temuerchaolu, Anco S., New conservation laws obtained directly from symmetry action on a known conservation law, J. Math. Anal. Appl., 2005, to appear.
  13. Bluman G., Doran-Wu P., The use of factors to discover potential symmetries and linearizations, Acta Appl. Math., 1995, V.41, 21-43.
  14. Bluman G., Cheviakov A., Framework for potential systems and nonlocal symmetries: Algorithmic approach, J. Math. Phys., 2005, to appear.
  15. Kumei S., Bluman G., When nonlinear differential equations are equivalent to linear differential equations, SIAM J. Appl. Math., 1982, V.42, 1157-1173.
  16. Bluman G., Kumei S., Symmetry-based algorithms to relate partial differential equations, I. Local symmetries, European J. Appl. Math., 1990, V.1, 189-216.
  17. Bluman G., Kumei S., Symmetry-based algorithms to relate partial differential equations, II. Linearization by nonlocal symmetries, European J. Appl. Math., 1990, V.1, 217-223.
  18. Anco S., Wolf T., Bluman G., Computational linearization of partial differential equations by conservation laws, Dept. of Math., Brock University, St. Catharines, ON, Canada, 2005, in preparation.
  19. Bluman G., Anco S., Symmetry and integration methods for differential equations, New York, Springer, 2002.
  20. Popovych R.O., Ivanova N.M., Hierarchy of conservation laws of diffusion-convection equations, J. Math. Phys., 2005, V.46, 043502, 22 pages.
  21. Stephani H., Differential equations: Their solution using symmetries, Cambridge University Press, Cambridge, 1989.
  22. Kingston G., Sophocleous C., Symmetries and form-preserving transformations of one-dimensional wave equations with dissipation, Int. J. Non-Linear Mech., 2001, V.36, 987-997.
  23. Bluman G., Temuerchaolu, Sahadevan R., Local and nonlocal symmetries for nonlinear telegraph equations, J. Math. Phys., 2005, V.46, 023505, 12 pages.
  24. Bluman G., Temeuerchaolu, Conservation laws for nonlinear telegraph equations, J. Math. Anal. Appl., 2005, V.310, 459-476.
  25. Bluman G., Temuerchaolu, Comparing symmetries and conservation laws of nonlinear telegraph equations, J. Math. Phys., 2005, V.46, 073513, 14 pages.

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