Presymplectic Lagrangian systems. I: The constraint algorithm and the equivalence theorem.

*(English)*Zbl 0414.58015##### MSC:

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |

58A05 | Differentiable manifolds, foundations |

58A10 | Differential forms in global analysis |

70Hxx | Hamiltonian and Lagrangian mechanics |

##### Keywords:

Legendre transformation; presymplectic manifolds; global presymplectic geometry; degenerate Lagrangian systems; Hamiltonian formulations of dynamical systems; geometric constraint algorithm##### References:

[1] | R. Abraham and J. Marsden , Foundations of Mechanics , Benjamin , New York , second edition, 1978 . MR 515141 | Zbl 0393.70001 · Zbl 0393.70001 |

[2] | J. Klein , Ann. Inst. Fourier ( Grenoble ), t. 12 , 1962 , p. 1 ; Symposia Mathematica XIV ( Rome Conference on Symplectic Manifolds ), 181 , 1973 . MR 215269 |

[3] | D.J. Simms and N.M.J. Woodhouse , Lectures on Geometric Quantization , Lecture Notes in Physics , t. 53 , Springer-Verlag , Berlin , 1976 . MR 672639 | Zbl 0343.53023 · Zbl 0343.53023 |

[4] | A nice summary is given in P.A.M. Dirac , Lectures on Quantum Mechanics , Belfer Graduate School of Science Monograph Series , t. 2 , 1964 . Several examples are presented in Hanson, Regge and Teitelboim , Accademia Nazionale dei Lincei ( Rome ), t. 22 , 1976 . |

[5] | 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 |

[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] | From the point of view of the constraint algorithm, the homogeneous case is trivial because E \equiv 0 (see section III). |

[8] | J.M. Nester and M.J. Gotay , Presymplectic Lagrangian Systems II: The Second-Order Equation Problem (in preparation). Numdam | Zbl 0453.58016 · Zbl 0453.58016 · numdam:AIHPA_1980__32_1_1_0 · eudml:76059 |

[9] | J. Sniatycki , Proc. 13th Biennial Seminar of the Canadian Math. Cong. , t. 2 , 1972 , p. 125 . MR 371202 | Zbl 0273.58003 · Zbl 0273.58003 |

[10] | Throughout this paper, we assume for simplicity that all physical systems under consideration are time-independent and that all relevant phasespaces are finite-dimensional; however, all of the theory developed in this paper can be applied when these restrictions are removed with little or no modification. For details concerning the infinite-dimensional case, see refs. [5] and [18]. |

[11] | C. Godbillon , Géométrie Différentielle et Mécunique Analytique , Hermann , Paris , 1969 . MR 242081 | Zbl 0174.24602 · Zbl 0174.24602 |

[12] | We herein establish some notation and terminology. All manifolds and maps appearing in this paper are assumed to be C\infty . We designate the natural pairing TM x T*M \rightarrow R by <|>. The symbol i denotes the interior product. Note that if \gamma is a p-form, and X1, ..., Xp are vectorfields, then i(X1) ... i(Xp)\gamma = \gamma (Xp, ... , X1). The symbol | N means restriction to the submanifold N . If j : N \rightarrow M is the inclusion, then we denote by \gamma | N the restriction of \gamma to N. Given a 2-form \Omega on M, we define the \Omega -orthogonal complement of TN in TM to be TN1 = Z\in TM such that \Omega (Z, Y) = 0 for all Y\in TN. Furthermore, we define ker \Omega = Y\in TM such that i(Y)\Omega = 0 . If f : M \rightarrow P is smooth, then we denote by T f or f* the derived mapping TM \rightarrow TP. We have ker T f = Y\in TM such that T f(Y) = 0 . |

[13] | For another definition of FL (which is logically independent of the almost tangent structure J), see ref. [1]. |

[14] | J.M. Nester , Invariant Derivation of the Euler-Lagrange Equations (in preparation). |

[15] | We assume that all of the Pl appearing in the algorithm are in fact imbedded submanifolds. Otherwise, one must resort to standard tricks, e. g., work locally where everything is manageable (see Section IV). |

[16] | In fact, there does not even exist a unique local Hamiltonian formalism corresponding to such a Lagrangian system, as, e. g., with L = 1/4v4 - 1/2v2 . |

[17] | In the following, TM1/1 denotes the \omega 1-orthogonal complement (see [12]). |

[18] | M.J. Gotay , Presymplectic Manifolds, Geometric Constraint Theory and the Dirac-Bergmann Theory of Constraints , Ph. D. Thesis , University of Maryland , 1979 . |

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