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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.) |

### Keywords:

presymplectic Lagrangian systems; second-order equation problem for degenerate Lagrangian systems; global presymplectic geometry### Citations:

Zbl 0414.58015
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\textit{M. J. Gotay} and \textit{J. M. Nester}, Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A 32, 1--13 (1980; Zbl 0453.58016)

### 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 |

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[5] | J.M. Nester , Invariant Derivation of the Euler-Lagrange Equations (unpublished).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

[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 |

[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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