Spaces of solutions of relativistic field theories with constraints

  • Jerrold E. Marsden
I. Session in Honour of Konrad Bleuler
Part of the Lecture Notes in Mathematics book series (LNM, volume 905)


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Arms, J., Linearization stability of the Einstein-Maxwell system, J. Math. Phys. 18, 830–3 (1977).MathSciNetCrossRefGoogle Scholar
  2. [2]
    Arms, J., Linearization stability of gravitational and gauge fields, J. Math. Phys. 20, 443–453 (1979).MathSciNetCrossRefGoogle Scholar
  3. [3]
    Arms, J., The structure of the solution set for the Yang-Mills equations (preprint) (1980).Google Scholar
  4. [4]
    Arms, J., Fischer, A., and Marsden, J., Une approche symplectique pour des théorèmes de decomposition en geométrie ou relativité générale, C. R. Acad. Sci. Paris 281, 517–520 (1975).MathSciNetMATHGoogle Scholar
  5. [5]
    Arms, J., and Marsden, J., The absence of Killing fields is necessary for linearization stability of Einstein's equations, Ind. Univ. Math. J. 28, 119–125 (1979).MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Arms, J., Marsden, J., and Moncrief, V., Bifurcations of momentum mappings, Comm. Math. Phys. 78 455–478 (1981).MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Arms, J., Marsden, J., and Moncrief, V., The structure of the space of solutions of Einstein's equations; II Many Killing fields, (in preparation) (1981).Google Scholar
  8. [8]
    Atiyah, M.F., Hitchin, N.J., and Singer, I.M., Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London, A 362, 425–461 (1978).MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Bao, D., Linearization Stability of Supergravity, Thesis, Berkeley (1982).Google Scholar
  10. [10]
    Brill, D., and Deser, S., Instability of closed spaces in general relativity, Comm. Math. Phys. 32, 291–304 (1973).MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Chernoff, P. and Marsden, J., Properties of Infinite Dimensional Hamiltonian Systems, Springer Lecture Notes in Math. No. 425 (1974).Google Scholar
  12. [12]
    Choquet-Bruhat, Y., and Deser, S., Stabilité initiale de l'espace temps de Minkowski, C. R. Acad. Sci. Paris 275, 1019–1027 (1972).MathSciNetGoogle Scholar
  13. [13]
    Choquet-Bruhat, Y., and Deser, S., On the stability of flat space. Ann. of Phys. 81, 165–168 (1973).MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Choquet-Bruhat, Y., Fischer, A, and Marsden, J., Maximal Hypersurfaces and Positivity of Mass, in Isolated Gravitating Systems and General Relativity, J. Ehlers ed., Italian Physical Society, 322–395 (1979).Google Scholar
  15. [15]
    Christodoulou, D. and O'Murchadha, N., The boost problem in general relativity, (preprint) (1980).Google Scholar
  16. [16]
    Coll, B. Sur la détermination, par des donnés de Cauchy, des champs de Killing admis par un espace-time, d'Einstein-Maxwell, C.R. Acad. Sci. (Paris), 281, 1109–12, 282, 247–250, J Math. Phys. 18, 1918–22 (1975)MathSciNetMATHGoogle Scholar
  17. [17]
    Cordero, P. and Teitelboim, C., Hamiltonian treatment of the spherically symmetric Einstein-Yang-Mills system, Ann. Phys. (N.Y.), 100, 607–31 (1976).MathSciNetCrossRefGoogle Scholar
  18. [18]
    D'Eath, P.D., On the existence of perturbed Robertson-Walker universes, Ann. Phys. (N.Y.) 98, 237–63 (1976).MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Deser, S., Covariant decomposition of symmetric tensors and the gravitational Cauchy problem, Ann. Inst. H. Poincaré VII, 149–188 (1967).Google Scholar
  20. [20]
    Ebin, D. and Marsden, J., Groups of Diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 102–163 (1970).MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    Fischer, A. and Marsden, J., The Einstein Evolution equations as a first-order symmetric hyperbolic quasilinear system, Commun. Math. Phys. 28, 1–38 (1972).MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    Fischer, A. and Marsden, J., The Einstein equations of evolution—a geometric approach, J. Math. Phys. 13, 546–68 (1972).MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    Fischer, A. and Marsden, J., Linearization stability of the Einstein equations, Bull. Am. Math. Soc. 79, 995–1001 (1973).MathSciNetCrossRefGoogle Scholar
  24. [24]
    Fischer, A. and Marsden, J., The initial value problem and the dynamical formulation of general relativity, in “General Relativity, An Einstein Centenary Survey, ed. S. Hawking and W. Israel, Cambridge, Ch. 4 (1979).Google Scholar
  25. [25]
    Fischer, A. and Marsden, J., Topics in the dynamics of general relativity, in Isolated gravitating systems in General Relativity, J. Ehlers, ed., Italian Physical Society, 322–395 (1979).Google Scholar
  26. [26]
    Fischer, A., Marsden, J., and Moncrief, V., The structure of the space of solutions of Einstein's equations, I: One Killing Field, Ann. Inst. H. Poincaré 33, 147–194 (1980).MathSciNetMATHGoogle Scholar
  27. [27]
    Garcia, P., The Poincaré-Cartan invariant in the calculus of variations, Symp. Math. 14, 219–246 (1974).Google Scholar
  28. [28]
    Garcia, P., (See his article in this volume) (1981).Google Scholar
  29. [29]
    Goldschmidt, H. and Sternberg, S., The Hamilton-Jacobi formalism in the calculus of variations, Ann. Inst. Fourier 23, 203–267 (1973).MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    Gotay, M., Nester, J., and Hinds, G., Pre-symplectic manifolds and the Dirac-Bergman theory of constraints, J. Math. Phys. 19, 2388–99 (1978).MathSciNetMATHCrossRefGoogle Scholar
  31. [31]
    Hansen, A., Regge, T., and Teitelboim, C., Constrained Hamiltonian Systems, Accademia Nazionale dei Lincei, Scuola Normale Superlore, Pisa (1976).Google Scholar
  32. [32]
    Hawking, S. and Ellis, G., The Large Scale Structure of Spacetime Cambridge University Press, (1973).Google Scholar
  33. [33]
    Hughes, T., Kato, T., and Marsden, J., Well-posed quasi-linear hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rat. Mech. An. 63, 273–294 (1977).MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    Kijowski, J. and Tulczyjew, W., A symplectic framework for field theories, Springer Lecture Notes in Physics No. 107 (1979).Google Scholar
  35. [35]
    Kuranishi, M., New proof for the existence of locally complete families of complex structures. Proc. of Conf. on Complex Analysis, (A. Aeppli et al., eds.), Springer (1975).Google Scholar
  36. [36]
    Marsden, J., Hamiltonian One Parameter Groups, Arch. Rat. Mech. An. 28 361–396 (1968).Google Scholar
  37. [37]
    Marsden, J., Lectures on Geometric Methods in Mathematical Physics, SIAM (1981).Google Scholar
  38. [38]
    Marsden, J. and Weinstein, A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5, 121–130 (1974).MathSciNetMATHCrossRefGoogle Scholar
  39. [39]
    Misner, C., Thorne, K., and Wheeler, J., Gravitation, Freeman, San Francisco, (1973).Google Scholar
  40. [40]
    Moncrief, V., Spacetime symmetries and linearization stability of the Einstein equations, J. Math. Phys. 16, 493–498 (1975).MathSciNetMATHCrossRefGoogle Scholar
  41. [41]
    Moncrief, V., Decompositions of gravitational perturbations, J. Math. Phys. 16, 1556–1560 (1975).MathSciNetCrossRefGoogle Scholar
  42. [42]
    Moncrief, V., Spacetime symmetries and linearization stability of the Einstein equations II, J. Math. Phys. 17, 1893–1902 (1976).MathSciNetCrossRefGoogle Scholar
  43. [43]
    Moncrief, V., Gauge symmetries of Yang-Mills fields, Ann. of Phys. 108, 387–400 (1977).MathSciNetCrossRefGoogle Scholar
  44. [44]
    Moncrief, V., Invariant states and quantized gravitational perturbations, Phys. Rev. D. 18, 983–989 (1978).MathSciNetCrossRefGoogle Scholar
  45. [45]
    Nelson, J. and Teitelboim, C., Hamiltonian for the Einstein-Dirac field, Phys. Lett. 69B, 81–84 (1977), and Ann. of Phys. (1977).Google Scholar
  46. [46]
    O'Murchadha, N. and York, J., Existence and uniqueness of solutions of the Hamiltonian constraint of general relativity on compact manifolds, J. Math. Phys. 14, 1551–1557 (1973).MathSciNetMATHCrossRefGoogle Scholar
  47. [47]
    Pilati, M., The canonical formulation of supergravity, Nuclear Physics B 132, 138–154 (1978).MathSciNetCrossRefGoogle Scholar
  48. [48]
    Regge, T. and Teitelboim, C., Role of surface integrals in the Hamiltonian formulation of general relativity, Ann. Phys. (N.Y.), 88, 286–318 (1974).MathSciNetMATHCrossRefGoogle Scholar
  49. [49]
    Segal, I., Differential operators in the manifold of solutions of nonlinear differential equations, J. Math. Pures. et Appl. XLIV, 71–132 (1965).Google Scholar
  50. [50]
    Segal, I., General properties of Yang-Mills fields on physical space, Nat. Acad. Sci. 75, 4638–4639 (1978).MATHCrossRefGoogle Scholar
  51. [51]
    Segal, I., The Cauchy problem for the Yang-Mills equations, J. Funct. An. 33, 175–194 (1979).MATHCrossRefGoogle Scholar
  52. [52]
    Sniatycki, J., On the canonical formulation of general relativity, Journées Relativistes de Caen, Soc. Math. de France and Proc. Camb. Phil. Soc. 68, 475–484 (1970).MathSciNetMATHCrossRefGoogle Scholar
  53. [53]
    Szczyryba, W., A symplectic structure of the set of Einstein metrics: a canonical formalism for general relativity, Comm. Math. Phys. 51, 163–182 (1976).MathSciNetCrossRefGoogle Scholar
  54. [54]
    Taub, A., Variational Principles in General Relativity, C.I.M.E. Bressanone, 206–300 (1970).Google Scholar
  55. [55]
    Weinstein, A., A Universal Phase Space for Particles in Yang-Mills fields, Lett. Math. Phys. 2, 417–420 (1979).CrossRefGoogle Scholar
  56. [56]
    York, J.W., Covariant decompositions of symmetric tensors in the theory of gravitation, Ann. Inst. H. Poincaré 21, 319–32 (1974).MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1982

Authors and Affiliations

  • Jerrold E. Marsden
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

Personalised recommendations