La Rivista del Nuovo Cimento (1978-1999)

, Volume 6, Issue 2, pp 1–62 | Cite as

Dynamics and symmetry for constrained systems: a geometrical analysis

  • G. Marmo
  • N. Mukunda
  • J. Samuel



field of real numbers


configuration space

TmQ, Tm*Q

tangent and cotangent spaces at m ε Q

TQ, T * Q

tangent and cotangent bundles of Q

T(Q), X(Q),X * (Q)

algebras of functions, vector fields, 1-forms

TΦ:TQ →TN:TN (Φ [cq])

[Φ o c]Φ(q), where [cq] is the equivalence class of curves tangent at qε Q and ΦQ → N is C∞


exterior derivative


Lie and inner derivative with respect to X ∈ X(Q)


Cartan 1-form on TQ

ω ℒ

d θ ℒ


canonical 2-form on T * Q

in → M

identification map of the submanifold N into M<┐>


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  1. [1]
    P. A. M. Dirac:Can. J. Math.,2, 129 (1950);Troc. R. Soc. London, Ser. A,246, 326 (1958).MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    J. L. Anderson andP. G. Bergman:Phys. Rev.,83, 1018 (1951);P. G. Bergman andI. Goldberg:Phys. Rev.,98, 531 (1955).ADSCrossRefMATHGoogle Scholar
  3. [3]
    For a comprehensive review, seeA. J. Hanson,T. Regge andC. Teitelboim:Constrained Hamiltonian Systems, Accademia Nazionale dei Lincei (Rome, 1976).Google Scholar
  4. [4]
    A recent account is provided byK. Sundermeyer:Constrained dynamics, inLecture Notes in Physics, Vol.169 (Berlin, 1982).Google Scholar
  5. [5]
    P. A. M. Dirac:Phys. Rev.,74, 817 (1948).MathSciNetADSCrossRefMATHGoogle Scholar
  6. [6]
    See, for instance,C. Rebbi:Phys. Bep. C,12, No. 1 (1974).Google Scholar
  7. [7]
    A. J. Hanson andT. Regge:Ann. Phys. (N. T.),87, 498 (1974).MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    N. Mukunda, H.Van Dam andL. C. Biedenharm:Phys. Rev. D,22, 1938 (1980).MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    Out of a vast literature, we mentionA. Komar:Phys. Rev. D),18, 1987 (1978);D. Dominici,J. Gomis andG. Longhi:Nuovo Cimento A,48, 257 (1978);Nuovo Cimento B,48, 152 (1978) ;F. Rohrlich:Ann. Phys. (N. Y.),117, 292 (1979) ;T. Takabatashi:Suppl. Prog. Theor. Phys.,67, 1 (1979);A. Kihlberg,R. Marnelius andN. Mukunda:Phys. Rev. D,23, 2201 (1981);N. Mukunda andE. C. G. Sudarshan:Phys. Rev. D,23, 2210 (1981);E. C. G. Sudarshan,N. Mukunda andJ. Goldberg:Phys. Rev. D,23, 2218 (1981).MathSciNetADSGoogle Scholar
  10. [10]
    In the language of differential geometry:H. P. Kunzle:Symposia Mathematica, Vol.14 (1974), p. 53;A. P. Balachandran, G. Marmo, N. Mukunda, J. Nilsson, A. Simoni, E. C. G. Sudarshan andF. Zaccaria:A unified geometrical approach to relativistic particle dynamics, J. Math. Phys. (N. Y.), in press.MathSciNetGoogle Scholar
  11. [11]
    H. P. Kunzle:Ann. Inst. Henry Poincaré, Ser. A,11, 393 (1969);J. Sniattcki:Ann. Inst. Henry Poincaré, Ser. A,20, 365 (1974);M. R. Menzio andW. M. Tulcztjew:Ann. Inst. Henry Poincaré, Ser. A,23, 349 (1978);A. Lichnerowicz:C. B. Acad. Sci. Paris, Ser. A,280, 523 (1975).MathSciNetGoogle Scholar
  12. [12]
    M. J. Gotat, J. M. Nester andG. Hinds:J. Math. Phys. (N. T.),19, 2388 (1978);M. J. Gotat andJ. M. Nester:Ann. Inst. Henry Poincaré, Ser. A, 30, 129 (1979);32, 1 (1980).ADSCrossRefGoogle Scholar
  13. [13]
    N. Mukunda:Ann. Phys. (N. T.),99, 408 (1976);Phys. Scr.,21, 783 (1980).MathSciNetADSCrossRefMATHGoogle Scholar
  14. [14]
    R. Abraham andJ. E. Marsden :Foundations of Mechanics (New York, N. Y., 1978).Google Scholar
  15. [15]
    G. Marmo, E. J. Saletan andA. Simoni:Nuovo Cimento B,50, 21 (1979).MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    G. Marmo, E. J. Saletan andA. Simoni:J. Math. Phys. (N. Y.),20, 856 (1979).MathSciNetADSCrossRefMATHGoogle Scholar

Copyright information

© Società Italiana di Fisica 1983

Authors and Affiliations

  • G. Marmo
    • 1
    • 2
  • N. Mukunda
    • 3
  • J. Samuel
    • 4
  1. 1.Sezione di NapoliIstituto Nazionale di Fisica NucleareItalia
  2. 2.Istituto di Fisica Teorica dell’ UniversitàNapoliItalia
  3. 3.Center for Theoretical Studies and Department of PhysicsIndian Institute of ScienceBangaloreIndia
  4. 4.Raman Research InstituteBangaloreIndia

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