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Presymplectic Lagrangian systems. II: The second-order equation problem. (English) Zbl 0453.58016

##### MSC:
 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 70H03 Lagrange’s equations 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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##### References:
 [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 , 1979 t p. 129 . Numdam | MR 535369 | Zbl 0414.58015 · Zbl 0414.58015 · numdam:AIHPA_1979__30_2_129_0 · eudml:76022 [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 569299 | Zbl 0438.58016 · Zbl 0438.58016 [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 [5] J.M. Nester , Invariant Derivation of the Euler-Lagrange Equations (unpublished).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 278586 | Zbl 0193.24901 · Zbl 0193.24901 · numdam:AIHPA_1969__11_4_393_0 · eudml:75647 [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 215269 [10] C. Godbillon , Géométrie Différentielle et Mécanique Analytique ( Hermann , Paris , 1969 ). MR 242081 | Zbl 0174.24602 · Zbl 0174.24602 [11] P. Rodrigues , C. R. Acad. Sci. Paris , A 281 , 1975 , p. 643 ; A 282 , 1976 , p. 1307 . Zbl 0312.53024 · Zbl 0312.53024 [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 506712 | Zbl 0418.58010 · Zbl 0418.58010 · doi:10.1063/1.523597 [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, \Omega , 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), \omega 1, dH1) by (L, \Omega , 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].
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