Skip to main content
SpringerLink
Log in

Necessary conditions for existence of non-degenerate Hamiltonian structures

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The necessary criteria are pointed out for the exisence of Hamiltonian and bi-Hamiltonian non-degenerate structures for a nonlinear system of partial differential equations of first order. The results are formulated in terms of the new invariants of the intrinsic geometry, introduced in this paper, connected with the Nijenhuis and Haantjes tensors of a (1,1) tensor field.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Riemann, G.F.B.: Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite. Abhandl. Konigl. Ges. Wiss. Gőttingen, v.8, 1860.

  2. Courant, R., Hilbert, D.: Methods of mathematical physics, v. II. New York: Interscience Publishers, 1962

    Google Scholar 

  3. Lax, P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math.7, 159–193 (1954)

    MATH  MathSciNet  Google Scholar 

  4. Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math.18, 697–715 (1965)

    MATH  MathSciNet  Google Scholar 

  5. Nijenhuis, A.:X n-1 -forming sets of eigenvectors. Proc. Kon. Ned. Akad. Amsterdam54, 200–212 (1951)

    MATH  MathSciNet  Google Scholar 

  6. Haantjes, J.: OnX m -forming sets of eigenvectors. Proc. Kon. Ned. Akad. Amsterdam58, 158–162 (1955)

    MATH  MathSciNet  Google Scholar 

  7. Dubrovin, B.A., Novikov, S.P.: On Poisson brackets of hydrodynamic type. Sov. Math. Dokl.30, 651–654 (1984)

    MATH  Google Scholar 

  8. Novikov, S.P.: The geometry of conservative systems of hydrodynamic type. The method of averaging for field-theoretical systems. Russ. Math. Surv.40, N4, 85–98 (1985)

    Article  MATH  Google Scholar 

  9. Dubrovin, B.A., Novikov, S.P.: Hydrodynamics of weakly deformed solition lattices. Differential geometry and Hamiltonian theory. Russ. Math. Surv.44, N6, 35–124 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  10. Magri, F.: A simple model of an integrable Hamiltonian system. J. Math. Phys.19, 1156–1162 (1978)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Bogoyavlenskij, O.I.: Existence of Riemann invariants and Hamiltonian structures. C.R. Math. Rep. Acad. Sci. Canada15, 143–148 (1993)

    MATH  MathSciNet  Google Scholar 

  12. Benney, D.J.: Some properties of long non-linear waves. Stud. Appl. Math.52, 45–50 (1973)

    MATH  Google Scholar 

  13. Whitham, G.B.: Non-linear dispersive waves. Proc. Royal Soc. Lond. Ser. A139, 283–291 (1965)

    Google Scholar 

  14. Ercolani, N., Forest, M.G., McLaughlin, D.W., Montgomery, R.: Hamiltonian structure of the modulation equations of a Sine-Gordon wavetrain. Duke Math. J.55, 949–983 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  15. Flaschka, H., Forest, M.G., McLaughlin, D.W.: Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation. Commun. Pure Appl. Math.33, 739–784 (1980)

    MATH  MathSciNet  ADS  Google Scholar 

  16. Stone, A.P.: Generalized conservation laws. Proc. of Am. Math. Soc.18, 868–873 (1967)

    Article  MATH  Google Scholar 

  17. Weyl, H.: The classcal groups, their invariants and representations. Princeton, NJ.: Princeton University Press, 1946

    Google Scholar 

  18. Eisenhart, L.P.: Riemannian geometry. Princeton, NJ: Princeton University Press, 1964

    Google Scholar 

  19. Tsarev, S.P.: On Poisson brackets and Hamiltonian systems of hydrodynamic type. Sov. Math. Dokl.31, 488–491 (1985)

    MATH  Google Scholar 

  20. Mokhov, O.I., Ferapontov, E.V.: Non-local Hamiltonian operators of hydrodynamic type related to metrices of constant curvature. Russ. Math. Surv.45, N2, 218–219 (1990)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Queen's University, K7L 3N6, Kingston, Canada

    Oleg I. Bogoyavlenskij

Authors
  1. Oleg I. Bogoyavlenskij
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by H. Araki

Supported by NSERC grant OGPIN 337

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bogoyavlenskij, O.I. Necessary conditions for existence of non-degenerate Hamiltonian structures. Commun.Math. Phys. 182, 253–289 (1996). https://doi.org/10.1007/BF02517890

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02517890

Keywords

Search

Navigation