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The Cohomology of Invariant Variational Bicomplexes

  • Ian M. Anderson
  • Juha Pohjanpelto

Abstract

Let π: EM be a fiber bundle and let Γ be an infinitesimal Lie transformation group acting on E. We announce various new results concerning the cohomology of the Γ invariant variational bicomplex (Ω Γ *,* (J (E)),d H ,d V ) and the associated Γ invariant Euler-Lagrange complex. As one application of our general theory, we completely solve the local invariant inverse problem of the calculus of variations for finite-dimensional infinitesimal Lie transformation groups.

Keywords

Vector Field Regular Point Source Form Horizontal Connection Continuous Cohomology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Anderson, I. M.: Natural variational principles on Riemannian manifolds, Ann. Math. 120 (1984), 329–370.zbMATHCrossRefGoogle Scholar
  2. 2.
    Anderson, I. M.: Introduction to the variational bicomplex, in M. Gotay, J. Marsden and V. Moncrief (eds), Mathematical Aspects of Classical Field Theory, Contemporary Mathemathics 132, Amer. Math Soc, Providence, 1992, pp. 51–73.Google Scholar
  3. 3.
    Anderson, I. M.: The Variational Bicomplex, Academic Press, Boston (to appear).Google Scholar
  4. 4.
    Bott, R.: Notes on Gel’fand-Fuks cohomology and characteristic classes (notes by M. Mostow and J. Perchik), Eleventh Holiday Symposium, New Mexico State Univ., Las Cruces, 1973.Google Scholar
  5. 5.
    Cheung, W. S.: Higher order conservation laws and a higher order Noether’s theorem, Adv. Appi. Math. 8 (1987), 446–485.zbMATHGoogle Scholar
  6. 6.
    David, D., Kamran, N., Levi, D., and Winternitz, R: Symmetry reduction for the Kadomtsev-Petviashvili equation using a loop algebra, J. Math. Phys. 27 (1986), 1225–1237.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Fuks, D. B.: Cohomology of Infinite Dimensional Lie Algebras, Consultants Bureau, New York, 1986.Google Scholar
  8. 8.
    Gromov, M.: Partial Differential Relations, Springer-Verlag, Berlin, 1986.zbMATHGoogle Scholar
  9. 9.
    Hilton, P J. and Stammbach, U.: A Course in Homological Algebra, Springer-Verlag, New York, 1971.zbMATHGoogle Scholar
  10. 10.
    Jackiw, R.: Topological investigations of quantized gauge theories, in A. S. Wrightman and P. A. Anderson (eds), Current Algebras and Anomalies, Princeton Series in Physics, Princeton University Press, Princeton, 1985, pp. 210–360.Google Scholar
  11. 11.
    Lévy-Leblond, J. M.: Group-theoretical foundations of classical mechanics: the Lagrangian gauge problem, Comm. Math. Phys. 12 (1969), 64–79.zbMATHCrossRefGoogle Scholar
  12. 12.
    López, M. C, Noriega, R. J., and Schifini, C. G.: The equivariant inverse problem and the uniqueness of the Yang-Mills equations, J. Math. Phys. 30 (1989), 2382–2387.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Marmo, G., Saletan, E. J., and Simoni, A.: On obtaining strictly invariant Lagrangians from gauge invariant Lagrangians, Nuovo Cim. 96B (1986), 159–163.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Marmo, G. and Morandi, G.: Inverse problem with symmetries and the appearance of coho-mologies in classical Lagrangian dynamics, Rep. Math Phys. 3 (1989), 389–410.MathSciNetGoogle Scholar
  15. 15.
    Marmo, G., Morandi, G., and Rubano, C: Symmetries in the Lagrangian and Hamiltonian formalism: the equivariant inverse problem, in B. Gruber and F. Iachelle (eds), Symmetries in Science 111, Plenum, New York, 1989, pp. 243–309.CrossRefGoogle Scholar
  16. 16.
    Olver, P. J.: Applications of Lie Groups to Differential Equations, Springer, New York, 1986.Google Scholar
  17. 17.
    Olver, P. J.: Differential invariants, Acta Appi. Math. 41 (1995), 271–284 (this issue).Google Scholar
  18. 18.
    Ovsiannikov, L. V: Group Analysis of Differential Equations, Academic Press, New York, 1982.zbMATHGoogle Scholar
  19. 19.
    Noriega, R. J. and Schifini, C. G.: The equivariant inverse problem and the Maxwell equations, J. Math. Phys. 28 (1987), 815–817.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Tsujishita, T.: On variation bicomplexes associated to differential equations, Osaka J. Math. 19 (1982), 311–363.MathSciNetzbMATHGoogle Scholar
  21. 21.
    Tulczyjew, W. M.: The Euler-Lagrange resolution, Lecture Notes in Math. 836, Springer-Verlag, New York, 1980, pp. 22–48.Google Scholar
  22. 22.
    Tulczyjew, W. M.: Cohomology of the Lagrange complex, Ann. Scuola. Norm. Sup. Pisa 14 (1987), 217–227.MathSciNetzbMATHGoogle Scholar
  23. 23.
    Whiston, G. S.: On the gauge variance of action functions under transformations on space-time, Internat. J. Theoret. Phys. 5 (1972), 391–401.CrossRefGoogle Scholar
  24. 24.
    Witten, E.: Global aspects of current algebra, Nuclear Phys. B 223 (1983), 422–432.MathSciNetCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Ian M. Anderson
    • 1
  • Juha Pohjanpelto
    • 2
  1. 1.Department of MathematicsUtah State UniversityLoganUSA
  2. 2.Department of MathematicsOregon State UniversityCorvallisUSA

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