Presymplectic lagrangian systems. II : the second-order equation problem

Mark J. Gotay; James M. Nester

Annales de l'I.H.P. Physique théorique (1980)

  • Volume: 32, Issue: 1, page 1-13
  • ISSN: 0246-0211

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Gotay, Mark J., and Nester, James M.. "Presymplectic lagrangian systems. II : the second-order equation problem." Annales de l'I.H.P. Physique théorique 32.1 (1980): 1-13. <http://eudml.org/doc/76059>.

@article{Gotay1980,
author = {Gotay, Mark J., Nester, James M.},
journal = {Annales de l'I.H.P. Physique théorique},
keywords = {presymplectic Lagrangian systems; second-order equation problem for degenerate Lagrangian systems; global presymplectic geometry},
language = {eng},
number = {1},
pages = {1-13},
publisher = {Gauthier-Villars},
title = {Presymplectic lagrangian systems. II : the second-order equation problem},
url = {http://eudml.org/doc/76059},
volume = {32},
year = {1980},
}

TY - JOUR
AU - Gotay, Mark J.
AU - Nester, James M.
TI - Presymplectic lagrangian systems. II : the second-order equation problem
JO - Annales de l'I.H.P. Physique théorique
PY - 1980
PB - Gauthier-Villars
VL - 32
IS - 1
SP - 1
EP - 13
LA - eng
KW - presymplectic Lagrangian systems; second-order equation problem for degenerate Lagrangian systems; global presymplectic geometry
UR - http://eudml.org/doc/76059
ER -

References

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  1. [1] M.J. Gotay and J.M. Nester, Presymplectic Lagrangian Systems I: The Constrain, Algorithm and the Equivalence Theorem. Ann. Inst. H. Poincaré, t. A 30, 1979t p. 129. Zbl0414.58015MR535369
  2. [2] M.J. Gotay and J.M. Nester, Presymplectic Hamilton and Lagrange Systems, Gauge Transformations and the Dirac Theory of Constraints, in Proc. of the VIIth Intl. Colloq. on Group Theoretical Methods in Physics, Austin. 1978,Lecture Notes in Physics. Springer-Verlag, Berlin, t. 94, 1979, p. 272. 
  3. [3] M.J. Gotay and J.M. Nester, Generalized Constraint Algorithm and Special Presymplectic Manifolds, to appear in the Proc. of the NSF-CBMS Regional Conference on Geometric Methods in Mathematical Physics, Lowell, 1979. Zbl0438.58016MR569299
  4. [4] M.J. Gotay, Presymplectic Manifolds, Geometric Constraint Theory and the Dirac–Bergmann Theory of Constraints, Dissertation, Univ. of Maryland, 1979 (unpu blished). 
  5. [5] J.M. Nester, Invariant Derivation of the Euler-Lagrange Equations (unpublished). 
  6. [6] H.P. Künzle, Ann. Inst. H. Poincaré, t. A 11, 1969, p. 393. Zbl0193.24901MR278586
  7. [7] For example, take L = (1 + y)v2x - zx2 + y on TQ = TR3. 
  8. [8] Throughout this paper, we assume for simplicity that all physical systems under consideration have a finite number of degrees of freedom; however, all of the theory developed in this paper can be applied when this restriction is removed with little or no modification. For details concerning the infinite-dimensional case, see references [3], [4] and [12]. 
  9. [9] J. Klein, Ann. Inst. Fourier (Grenoble), t. 12, 1962, p. 1; Symposia Mathematica XIV (Rome Conference on Symplectic Manifolds), 1973, p. 181. MR215269
  10. [10] C. Godbillon, Géométrie Différentielle et Mécanique Analytique (Hermann, Paris, 1969). Zbl0174.24602MR242081
  11. [11] P. Rodrigues, C. R. Acad. Sci. Paris, A 281, 1975, p. 643 ; A 282, 1976, p. 1307. Zbl0312.53024
  12. [12] M.J. Gotay, J.M. Nester and G. Hinds, Presymplectic Manifolds and the Dirac–Bergmann Theory of Constraints. J. Math. Phys., t. 19, 1978, p. 2388. Zbl0418.58010MR506712
  13. [13] Elsewhere [3] we have developed a technique which will construct such an S—if it exists—for a completely general Lagrangian canonical system. However, the corresponding second-order equation X on S need not be smooth if (TQ, Ω, P) is not admissible. 
  14. [14] The requirement of admissibility is slightly weaker than that of almost regularity, cf. [1]. 
  15. [15] This is the case, e. g., in electromagnetism, cf. [4]. 
  16. [16] Nonetheless, by utilizing the technique alluded to in [13], it is possible to construct a unique maximal submanifold S' with the desired properties for any Lagrangian system whatsoever. However, unless the existence of S' actually follows from the Second-Order Equation Theorem, one is guaranteed neither that S' will be nonempty nor that the associated second-order equation X on S' will be smooth. 
  17. [17] With regard to the constructions of reference [1], one is effectively replacing « almost regular» by « admissible » and (FL(TQ), ω1, dH1) by (L, Ω, d'E). 
  18. [18] This proposition has the following useful corollary: if a solution of (3.5) is globally a second-order equation (i. e. (3.2) is satisfied on all of P), then it is not semi-prolongable, cf. [15]. 

Citations in EuDML Documents

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  1. Mark J. Gotay, James M. Nester, Presymplectic lagrangian systems. I : the constraint algorithm and the equivalence theorem
  2. Sergio de Filippo, Giovanni Landi, Giuseppe Marmo, Gaetano Vilasi, Tensor fields defining a tangent bundle structure
  3. F. Barone, R. Grassini, G. Mendella, A unified approach to constrained mechanical systems as implicit differential equations
  4. Monika Havelková, A geometric analysis of dynamical systems with singular Lagrangians
  5. Monika Havelková, Symmetries of a dynamical system represented by singular Lagrangians
  6. Manuel De León, Paulo R. Rodrigues, Dynamical connections and non-autonomous lagrangian systems
  7. M. C. Muñoz Lecanda, N. Roman Roy, Lagrangian theory for presymplectic systems

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