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

Gotay, Mark J.; Nester, James M.

Annales de l'institut Henri Poincaré. Section A, Physique Théorique, Tome 32 (1980) no. 1, pp. 1-13.

[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. | Numdam | MR | Zbl

[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] 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. | MR | Zbl

[4] M.J. Gotay, Presymplectic Manifolds, Geometric Constraint Theory and the Dirac-Bergmann Theory of Constraints, Dissertation, Univ. of Maryland, 1979 (unpu blished).

[5] J.M. Nester, Invariant Derivation of the Euler-Lagrange Equations (unpublished).

[6] H.P. Künzle, Ann. Inst. H. Poincaré, t. A 11, 1969, p. 393. | Numdam | MR | Zbl

[7] For example, take L = (1 + y)v2x - zx2 + y on TQ = TR3.

[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] J. Klein, Ann. Inst. Fourier (Grenoble), t. 12, 1962, p. 1; Symposia Mathematica XIV (Rome Conference on Symplectic Manifolds), 1973, p. 181. | MR

[10] C. Godbillon, Géométrie Différentielle et Mécanique Analytique (Hermann, Paris, 1969). | MR | Zbl

[11] P. Rodrigues, C. R. Acad. Sci. Paris, A 281, 1975, p. 643 ; A 282, 1976, p. 1307. | Zbl

[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. | MR | Zbl

[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] The requirement of admissibility is slightly weaker than that of almost regularity, cf. [1].

[15] This is the case, e. g., in electromagnetism, cf. [4].

[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] 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] 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].