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On the momentum mapping in field theory

  • Yvette Kosmann-Schwarzbach
II. Momentum Mappings And Invariants
Part of the Lecture Notes in Mathematics book series (LNM, volume 1139)

Keywords

Vector Bundle Hamiltonian System Cotangent Bundle Momentum Mapping Legendre Transformation 
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References

  1. [1]
    R. Abraham and J.E. Marsden, Foundations of Mechanics, 2 ed., Benjamin, Reading, Mass., 1978.zbMATHGoogle Scholar
  2. [2]
    V. Aldaya and J.A. de Azcárraga, Variational principles on r-th order jets of fiber bundles in field theory, J. Math. Phys. 19 (1978) 1869–1875.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    V. Aldaya and J.A. de Azcárraga, Geometric formulation of classical mechanics and field theory, Riv. Nuovo Cimento 3 (10) (1980).Google Scholar
  4. [4]
    R.L. Anderson and N.H. Ibragimov, Lie-Bäcklund Transformations in Applications, SIAM, Philadelphia 1979.CrossRefzbMATHGoogle Scholar
  5. [5]
    S. Benenti, M. Francaviglia and A. Lichnerowicz, eds., Modern developments in analytical mechanics, Proc. IUTAM — ISIMM Symposium (Torino 1982), Suppl. Atti Acad. Sc. Torino, Turin, 1983.Google Scholar
  6. [6]
    P.R. Chernoff and J.E. Marsden, Properties of infinite-dimensional Hamiltonian systems, Lect. Notes Math. 425, Springer-Verlag, Berlin 1974.CrossRefzbMATHGoogle Scholar
  7. [7]
    P.L. García, The Poincaré-Cartan invariant in the calculus of variations, Symposia Mathematica 14, Acad. Press, London 1974.zbMATHGoogle Scholar
  8. [8]
    I.M. Gel'fand and L.A. Dikii, Asymptotic behavior of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-de Vries equation, Russian Math. Surveys 30 (1975) 77–113.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    I.M. Gel'fand and I.Ya. Dorfman, Hamiltonian operators and algebraic structures related to them, Funct. Anal. Appl. 13 (1979) 248–262.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    I.M. Gel'fand and I. Ya. Dorfman, The Schouten bracket and Hamiltonian operators, Funct. Anal. Appl. 14 (1980) 223–226.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    H. Goldschmidt and S. Sternberg, The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier, Grenoble 23 (1973) 203–267.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    F. Guil Guerrero and L. Martinez Alonso, Generalized variational derivatives in field theory, J. Phys. A 13 (1980) 698–700.CrossRefzbMATHGoogle Scholar
  13. [13]
    C. Itzykson and J.B. Zuber, Quantum field theory, McGraw Hill, New York 1980.zbMATHGoogle Scholar
  14. [14]
    T. Iwai, Symmetry of vector wave equations dealt with in Hamiltonian formalism, Tensor N.S. 35 (1981) 205–215.MathSciNetzbMATHGoogle Scholar
  15. [15]
    J. Kijowski and W.M. Tulczyjew, A symplectic framework for field theories, Lect. Notes Physics 107, Springer-Verlag, Berlin 1979.CrossRefzbMATHGoogle Scholar
  16. [16]
    I. Kolář, Lie derivatives and higher order Lagrangians, in Proc. Conf. Diff. Geom. and Appl. (Prague 1980), Univ. Karlova, Prague 1981.zbMATHGoogle Scholar
  17. [17]
    I. Kolář, On the second tangent bundle and generalized Lie derivatives, Tensor N.S. 38 (1982) 98–102.MathSciNetzbMATHGoogle Scholar
  18. [18]
    Y. Kosmann-Schwarzbach, Vector fields and generalized vector fields on fibered manifolds, in Lect. Notes Math. 792, Springer-Verlag, Berlin 1980.zbMATHGoogle Scholar
  19. [19]
    Y. Kosmann-Schwarzbach, Hamiltonian systems on fibered manifolds, Lett. Math. Phys. 5 (1981) 229–237.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    D. Krupka, A geometric theory of ordinary first-order variational problems in fibered manifolds, I and II, J. Math. Anal. Appl. 49 (1975) 180–206 and 469–476.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    S. Kumei, On the relationship between conservation laws and invariance groups of nonlinear field equations in Hamilton's canonical form, J. Math. Phys. 19 (1978) 195–199.MathSciNetCrossRefGoogle Scholar
  22. [22]
    B.A. Kupershmidt, Lagrangian formalism in variational calculus, Funct. Anal. Appl. 10 (1976) 147–149.MathSciNetCrossRefGoogle Scholar
  23. [23]
    B.A. Kupershmidt, Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalism, in Lect. Notes Math. 775, G. Kaiser and J.E. Marsden, eds., Springer-Verlag, Berlin 1980.Google Scholar
  24. [24]
    B.A. Kupershmidt and G. Wilson, Modifying Lax equations and the second Hamiltonian structure, Invent. Math. 62 (1981) 403–436.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    P. Libermann et Ch.-M. Marle, Mécanique analytique et géométrie symplectique, Publ. Paris 7, à paraître; English transl. Reidel, to appear.Google Scholar
  26. [26]
    A. Lichnerowicz, Variétés de Poisson et feuilletages, Ann. Fac. Sc. Toulouse 4 (1982) 195–262.Google Scholar
  27. [27]
    Yu.I. Manin, Algebraic aspects of nonlinear differential equations, J. Soviet Math. 11 (1979) 1–122.CrossRefzbMATHGoogle Scholar
  28. [28]
    Ch.-M. Marle, Symplectic manifolds, dynamical groups and Hamiltonian mechanics, in Differential Geometry and Relativity, M. Cahen and M. Flato, eds., Reidel, Dordrecht 1976.Google Scholar
  29. [29]
    Ch.-M. Marle, Moment de l'action hamiltonienne d'un groupe de Lie, quelques propriétés, in [31]..Google Scholar
  30. [30]
    J.E. Marsden, A. Weinstein, T. Ratiu, R. Schmid and R.G. Spencer, Hamiltonian systems with symmetry, coadjoint orbits and plasma physics, in [5]..Google Scholar
  31. [31]
    M. Modugno, ed., Geometry and Physics (Florence 1982), Pitagora Editrice, Bologna 1983.zbMATHGoogle Scholar
  32. [32]
    E. Noether, Invariante Variationsprobleme, Nachr. Kön. Gesell. Wissen. Göttingen, Math. Phys. Kl. (1918) 235–257.Google Scholar
  33. [33]
    P.J. Olver, Applications of Lie groups to differential equations, Oxford University Lecture Notes, 1980 (to appear in Springer-Verlag Graduate Texts in Math. Series).Google Scholar
  34. [34]
    P.J. Olver, On the Hamiltonian structure of evolution equations, Math. Proc. Camb. Phil Soc. 88 (1980) 71–88.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    W.F. Shadwick, The Hamilton-Cartan formalism for r-th order Lagrangians and the integrability of the KdV and modified KdV equations, Lett. Math. Phys. 5 (1981) 137–141 (Erratum ibid. 6 (1982) 241).MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    W.F. Shadwick, The Hamiltonian structure associated to evolution-type Lagrangians, Lett. Math. Phys. 6 (1982) 271–276.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    J. Sńiatycki, On the geometric structure of classical field theory in Lagrangian formulation, Proc. Camb. Phil. Soc. 68 (1970) 475–484.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    J.M. Souriau, Structure des systèmes dynamiques, Dunod, Paris 1970.zbMATHGoogle Scholar
  39. [39]
    F. Takens, Symmetries, conservation laws and variational principles, Lect. Notes Math. 597, Springer-Verlag, Berlin 1977.zbMATHGoogle Scholar
  40. [40]
    A. Trautman, Noether equations and conservation laws, Comm. Math. Phys. 6 (1967) 248–261.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    A. Trautman, Invariance of Lagrangian systems, in General Relativity, L. O'Raifeartaigh, ed., Clarendon Press, Oxford 1972.Google Scholar
  42. [42]
    Tu Gui-zhang, Infinitesimal canonical transformations of generalized Hamiltonian equations, J. Phys. A 15 (1982) 277–285.MathSciNetCrossRefGoogle Scholar
  43. [43]
    A.M. Vinogradov, On the algebro-geometric foundations of Lagrangian field theory, Soviet Math. Dokl. 18 (1977) 1200–1204.zbMATHGoogle Scholar
  44. [44]
    A.M. Vinogradov, Hamilton structures in field theory, Soviet Math. Dokl. 19 (1978) 790–794.zbMATHGoogle Scholar
  45. [45]
    A.M. Vinogradov, Local symmetries and conservation laws, Acta Appl. Math. 2 (1984) 21–78.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Yvette Kosmann-Schwarzbach
    • 1
  1. 1.U.E.R. de MathématiquesUniversité de Lille IVilleneuve d'AscqFrance

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