Presymplectic Hamilton and Lagrange systems, Gauge transformations and the Dirac theory of constraints

  • Mark J. Gotay
  • James M. Nester
Symplectic Structure and Geometric Quantization
Part of the Lecture Notes in Physics book series (LNP, volume 94)


Hamiltonian System Lagrangian System True Degree Dirac Theory Reduce Phase Space 


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  1. 1.a
    R. Abraham and J. Marsden, Foundations of Mechanics (2nd Edition) Benjamin 1978.Google Scholar
  2. 1.b
    P. Chernoff and J. Marsden, Properties of Infinite Dimensional Hamiltonian Systems, Lecture notes in Math #425, Springer-Verlag 1974.Google Scholar
  3. 2.
    H.P. Künzle, J. Math. Phys. 13, 739 (1972).Google Scholar
  4. 3.
    M. Gotay, J. Nester, G. Hinds, “Presymplectic Manifolds and the Dirac-Bergmann Theory of Constraints”, to appear J. Math. Phys. (1978).Google Scholar
  5. 4.
    We assume here and throughout that quotients and constraints yield sufficiently nice manifolds. For many interesting systems this is not true-one must then cut the system up into nice pieces. See refs. 8 and 12.Google Scholar
  6. 5.
    Actually this is a particular application of a general integrability algorithm that we have developed for systems of equations.Google Scholar
  7. 6.
    P.A.M. Dirac, Lectures on Quantum Mechanics, Academic Press 1965.Google Scholar
  8. 7.
    A. Hanson, T. Regge, C. Teitelboim, “Constrained Hamiltonian Systems”, Accademia Nazionale dei Lincei #22, Rome (1976).Google Scholar
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  14. 13.
    J. Klein, Symposia Mathematica 14, 181 (1974).Google Scholar
  15. 14.
    This generalizes Śniatycki's definition.8 Google Scholar
  16. 15.
    Gotay and J. Nester, “Presymplectic Lagrangian Systems I: The Constraint Algorithm and the Equivalence Theorem”, submitted to Ann. Inst. Henri Poincaré.Google Scholar
  17. 16.
    Nester, “An Invariant Derivation of the Euler Lagrange Equations”, in preparation.Google Scholar
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    If L is degenerate (7) is not a consequence of (6).Google Scholar
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  20. 19.
    J. Nester and M. Gotay, “Presymplectic Lagrangian Systems II: The Second Order Equation Problem”, in preparation for Ann. Inst. Henri Poincaré.Google Scholar
  21. 20.
    See ref. 5, pp. 23,24.Google Scholar
  22. 21.
    J. Nester and M. Gotay, “Presymplectic Hamilton Equations: Gauge Transformations, and the Reduced Phase Space”, in preparation.Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Mark J. Gotay
    • 1
  • James M. Nester
    • 2
  1. 1.Center for Theoretical PhysicsUniversity of MarylandCollege Park
  2. 2.Department of PhysicsUniversity of SaskatchewanSaskatoonCanada

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