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

SIGMA 7 (2011), 063, 18 pages      arXiv:1101.5375      https://doi.org/10.3842/SIGMA.2011.063
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

Balance Systems and the Variational Bicomplex

Serge Preston
Department of Mathematics and Statistics, Portland State University, Portland, OR, 97207-0751, USA

Received January 27, 2011, in final form June 30, 2011; Published online July 09, 2011

Abstract
In this work we show that the systems of balance equations (balance systems) of continuum thermodynamics occupy a natural place in the variational bicomplex formalism. We apply the vertical homotopy decomposition to get a local splitting (in a convenient domain) of a general balance system as the sum of a Lagrangian part and a complemental ''pure non-Lagrangian'' balance system. In the case when derivatives of the dynamical fields do not enter the constitutive relations of the balance system, the ''pure non-Lagrangian'' systems coincide with the systems introduced by S. Godunov [Soviet Math. Dokl. 2 (1961), 947-948] and, later, asserted as the canonical hyperbolic form of balance systems in [Müller I., Ruggeri T., Rational extended thermodynamics, 2nd ed., Springer Tracts in Natural Philosophy, Vol. 37, Springer-Verlag, New York, 1998].

Key words: variational bicomplex; balance equations.

pdf (475 kb)   tex (22 kb)

References

  1. Anderson I., The variational bicomplex, Preprint, Utah State University, 2003.
  2. Bocharov A.V., Chetverikov V.N., Duzhin S.V., Khor'kova N.G., Krasil'shchik I.S., Samokhin A.V., Torkhov Yu.N., Verbovetsky A.M., Vinogradov A.M., Symmetries and conservation laws for differential equations of mathematical physics, American Mathematical Society, Providence, RI, 1999.
  3. Campos C., de Leon M., Diego D., Vankerschaver J. Unambiguous formalism for higher-order Lagrangian field theories, arXiv:0906.0389.
  4. Fatibene L., Francaviglia M., Natural and gauge natural formalism for classical field theories. A geometric perspective including spinors and gauge theories, Kluwer Academic Publishers, Dordrecht, 2003.
  5. Godunov S.K., An interesting class of quasilinear systems, Soviet Math. Dokl. 2 (1961), 947-948.
  6. Godunov S.K., Gordienko V.M., The simplest galilean-invariant and thermodynamically consistant conservative laws, J. Appl. Mech. Tech. Phys. 43 (2002), 1-12.
  7. Godunov S.K., Structure of thermodynamically compatible systems, Appendix to Godunov S.K., Romenskii E.I., Elements of continuum mechanics and conservation laws, Kluwer Academic/Plenum Publishers, New York, 2003.
  8. Kolár I., Michor P., Slovák J., Natural operations in differential geometry, Springer-Verlag, Berlin, 1993.
  9. Krupka D., Global variational theory in fibred spaces, in Handbook of Global Analysis, Editors D. Krupka and D. Saunders, Elsevier Sci. B.V., Amsterdam, 2008, 773-836.
  10. Marsden J.E., Hughes T.J.R., Mathematical foundations of elasticity, Dover Publications, Inc., New York, 1994.
  11. Müller I., Ruggeri T., Rational extended thermodynamics, 2nd ed., Springer Tracts in Natural Philosophy, Vol. 37, Springer-Verlag, New York, 1998.
  12. Olver P.J., Applications of Lie groups to differential equations, 2nd ed., Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1993.
  13. Preston S., Variational theory of balance systems, in Proceedings of Conf. "Differential Geometry and its Applications in Honour of Leonhard Euler" (August 27-31, 2007, Olomouc), Editors O. Kowalski, D. Krupka, O. Krupková and J. Slovák, World Sci. Publ., Hackensack, NJ, 2008, 675-688, arXiv:0806.4636.
  14. Preston S., Geometrical theory of balance systems, Int. J. Geom. Methods Mod. Phys. 7 (2010), 745-795.
  15. Preston S., Forms of Lepage type and the balance systems, Differential Geom. Appl., to appear.
  16. Saunders D., The geometry of jet bundles, London Mathematical Society Lecture Note Series, Vol. 142, Cambridge University Press, Cambridge, 1989.
  17. Senashov S.I., Vinogradov A.M., Symmetries and conservation laws of 2-dimensional ideal plasticity, Proc. Edinburgh Math. Soc. (2) 31 (1988), 415-439.
  18. Serre D., Systems of conservation laws. I. Hyperbolicity, entropies, shock waves, Cambridge University Press, Cambridge, 1999.
  19. Sieniutycz S., Conservation laws in variational thermohydrodynamics, Mathematics and its Applications, Vol. 279, Kluwer Academic Publishers Group, Dordrecht, 1994.
  20. Sieniutycz S., Farkas S., Variational and extremum principles in macroscopic systems, Elsevier, Amsterdam, 2005.

Previous article   Next article   Contents of Volume 7 (2011)