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Keywords:
fibred manifold; jet space; infinite order jet space; variational bicomplex; variational sequence 483504
fibred manifold; jet space; infinite order jet space; variational bicomplex; variational sequence 483504
Summary:
The theory of variational bicomplexes is a natural geometrical setting for the calculus of variations on a fibred manifold. It is a well–established theory although not spread out very much among theoretical and mathematical physicists. Here, we present a new approach to infinite order variational bicomplexes based upon the finite order approach due to Krupka. In this approach the information related to the order of jets is lost, but we have a considerable simplification both in the exposition and in the computations. We think that our infinite order approach could be easily applied in concrete situations, due to the conceptual simplicity of the scheme.
References:
[1] Anderson I. M., Duchamp T.: On the existence of global variational principles. Amer. Math. J. 102 (1980), 781-868. MR 0590637 | Zbl 0454.58021
[2] Bauderon M.: Le problème inverse du calcul des variations. Ann. de l’I.H.P. 36, n. 2 (1982), 159-179. MR 0662883 | Zbl 0519.58027
[3] Bott R., Tu L. W.: Differential Forms in Algebraic Topology. GTM 82 Springer–Verlag, Berlin, 1982. MR 0658304 | Zbl 0496.55001
[4] Dedecker P., Tulczyjew W. M.: Spectral sequences and the inverse problem of the calculus of variations. In Internat. Coll. on Diff. Geom. Methods in Math. Phys., Aix–en–Provence, 1979; Lecture Notes in Mathematics 836 Springer–Verlag, Berlin, 1980, 498-503. MR 0607719
[5] Ferraris M., Francaviglia M.: Global Formalism in Higher Order Calculus of Variations. Diff. Geom. and its Appl., Part II, Proc. of the Conf. University J. E. Purkyně, Brno, 1984, 93-117. MR 0793201
[6] Greub W.: Multilinear Algebra. Springer–Verlag, 1978. MR 0504976 | Zbl 0387.15001
[7] Kolář I.: A geometrical version of the higher order Hamilton formalism in fibred manifolds. Jour. Geom. Phys. 1, n. 2 (1984), 127-137. MR 0794983
[8] Kolář I., Vitolo R.: On the Helmholtz operator for Euler morphisms. preprint 1997. MR 2006065
[9] Krupka D.: Variational sequences on finite order jet spaces. Diff. Geom. and its Appl., Proc. of the Conf. World Scientific, New York, 1990, 236-254. MR 1062026 | Zbl 0813.58014
[10] Krupka D.: Topics in the calculus of variations: finite order variational sequences. Diff. Geom. and its Appl., Proc. of the Conf., Opava (Czech Republic), (1993) 473-495. MR 1255563 | Zbl 0811.58018
[11] Kuperschmidt B. A.: Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalism. Lecture Notes in Math. 775: Geometric Methods in Mathematical Physics, Springer, Berlin, (1980), 162-218. MR 0569303
[12] Mangiarotti L., Modugno M.: Fibered Spaces, Jet Spaces and Connections for Field Theories. Int. Meet. on Geometry and Physics, Proc. of the Conf. Pitagora Editrice, Bologna, 1983, 135-165. MR 0760841 | Zbl 0539.53026
[13] Modugno M., Vitolo R.: Quantum connection and Poincaré–Cartan form. Conference in honour of A. Lichnerowicz, Frascati, ottobre 1995; ed. G. Ferrarese, Pitagora, Bologna.
[14] Olver P. J., Shakiban C.: A Resolution of the Euler Operator. Proc. Am. Math. Soc. 69 (1978), 223-229. MR 0486822 | Zbl 0395.49002
[15] Saunders D. J.: The Geometry of Jet Bundles. Cambridge Univ. Press, 1989. MR 0989588 | Zbl 0665.58002
[16] Takens F.: A global version of the inverse problem of the calculus of variations. J. Diff. Geom. 14 (1979), 543-562. MR 0600611 | Zbl 0463.58015
[17] Tulczyjew W. M.: The Lagrange Complex. Bull. Soc. Math. France 105 (1977), 419-431. MR 0494272 | Zbl 0408.58020
[18] Tulczyjew W. M.: The Euler-Lagrange Resolution. Internat. Coll. on Diff. Geom. Methods in Math. Phys., Aix–en–Provence, 1979; Lecture Notes in Mathematics 836 Springer–Verlag, Berlin, 1980, 22-48. MR 0607685
[19] Vinogradov A. M.: On the algebro-geometric foundations of Lagrangian field theory. Soviet Math. Dokl. 18 (1977), 1200-1204. MR 0501142 | Zbl 0403.58005
[20] Vinogradov A. M.: A spectral sequence associated with a non-linear differential equation, and algebro–geometric foundations of Lagrangian field theory with constraints. Soviet Math. Dokl. 19 (1978), 144-148.
[21] Vitolo R.: Finite order Lagrangian bicomplexes. Math. Proc. of the Camb. Phil. Soc., to appear 124 n. 3, 1998.
[22] Vitolo R.: On different geometric formulations of Lagrangian formalism. preprint 1997, to appear on Diff. Geom. and Appl. MR 1692446
[23] Wells R. O.: Differential Analysis on Complex Manifolds. GTM 65 Springer–Verlag, Berlin, 1980. MR 0608414 | Zbl 0435.32004
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