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Symmetries in finite order variational sequences. (English) Zbl 1006.58014
Summary: We refer to Krupka’s variational sequence, i.e. the quotient of the de Rham sequence on a finite order jet space with respect to a ‘variationally trivial’ subsequence. Among the morphisms of the variational sequence there are the Euler-Lagrange operator and the Helmholtz operator.
In this note we show that the Lie derivative operator passes to the quotient in the variational sequence. Then we define the variational Lie derivative as an operator on the sheaves of the variational sequence. Explicit representations of this operator give us some abstract versions of Noether’s theorems, which can be interpreted in terms of conserved currents for Lagrangians and Euler-Lagrange morphisms.

58E30 Variational principles in infinite-dimensional spaces
58A12 de Rham theory in global analysis
58A20 Jets in global analysis
58J10 Differential complexes
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[1] I. M. Anderson and T. Duchamp: On the existence of global variational principles. Amer. Math. J. 102 (1980), 781-868. · Zbl 0454.58021
[2] L. Fatibene, M. Francaviglia and M. Palese: Conservation laws and variational sequences in gauge-natural theories. Math. Proc. Cambridge Phil. Soc. 130 (2001), 555-569. · Zbl 0988.58006
[3] M. Ferraris: Fibered connections and global Poincaré-Cartan forms in higher-order calculus of variations. Proc. Diff. Geom. and its Appl. (Nové Město na Moravě, 1983), D. Krupka (ed.), J. E. Purkyně University, Brno, 1984, pp. 61-91. · Zbl 0564.53013
[4] M. Ferraris and M. Francaviglia: The Lagrangian approach to conserved quantities in general relativity. Mechanics, Analysis and Geometry: 200 Years after Lagrange, M. Francaviglia (ed.), Elsevier Science Publishers B. V., Amsterdam, 1991, pp. 451-488.
[5] M. Francaviglia, M. Palese and R. Vitolo: Superpotentials in variational sequences. Proc. VII Conf. Diff. Geom. and Appl., Satellite Conf. of ICM in Berlin (Brno 1998), I. Kolář et al. (eds.), Masaryk University, Brno, 1999, pp. 469-480. · Zbl 0953.58015
[6] P. L. Garcia and J. Muñoz: On the geometrical structure of higher order variational calculus. Proc. IUTAM-ISIMM Symp. on Modern Developments in Anal. Mech. (Torino, 1982), S. Benenti, M. Francaviglia and A. Lichnerowicz (eds.), Tecnoprint, Bologna, 1983, pp. 127-147.
[7] I. Kolář, P. W. Michor and J. Slovák: Natural Operations in Differential Geometry. Springer-Verlag, New York, 1993. · Zbl 0782.53013
[8] I. Kolář: Lie derivatives and higher order Lagrangians. Proc. Diff. Geom. and its Appl. (Nové Město na Moravě, 1980), O. Kowalski (ed.), Univerzita Karlova, Praha, 1981, pp. 117-123.
[9] I. Kolář: A geometrical version of the higher order hamilton formalism in fibred manifolds. J. Geom. Phys. 1 (1984), 127-137. · Zbl 0595.58016
[10] D. Krupka: Some geometric aspects of variational problems in fibred manifolds. Folia Fac. Sci. Nat. UJEP Brunensis 14, J. E. Purkyně Univ., Brno (1973), 1-65.
[11] D. Krupka: Variational sequences on finite order jet spaces. Proc. Diff. Geom. and its Appl. (Brno, Czech Republic, 1989), J. Janyška, D. Krupka (eds.), World Scientific, Singapore, 1990, pp. 236-254. · Zbl 0813.58014
[12] D. Krupka: Topics in the calculus of variations: Finite order variational sequences. Proc. Diff. Geom. and its Appl, Opava, 1993, pp. 473-495. · Zbl 0811.58018
[13] D. Krupka and A. Trautman: General invariance of Lagrangian structures. Bull. Acad. Polon. Sci., Math. Astr. Phys. 22 (1974), 207-211. · Zbl 0278.49044
[14] L. Mangiarotti and M. Modugno: Fibered spaces, jet spaces and connections for field theories. Proc. Int. Meet. on Geom. and Phys., Pitagora Editrice, Bologna, 1983, pp. 135-165. · Zbl 0539.53026
[15] J. Novotný: Modern methods of differential geometry and the conservation laws problem. Folia Fac. Sci. Nat. UJEP Brunensis (Physica) 19 (1974), 1-55.
[16] D. J. Saunders: The Geometry of Jet Bundles. Cambridge Univ. Press, Cambridge, 1989. · Zbl 0665.58002
[17] F. Takens: A global version of the inverse problem of the calculus of variations. J. Diff. Geom. 14 (1979), 543-562. · Zbl 0463.58015
[18] A. Trautman: Noether equations and conservation laws. Comm. Math. Phys. 6 (1967), 248-261. · Zbl 0172.27803
[19] A. Trautman: A metaphysical remark on variational principles. Acta Phys. Pol. B 27 (1996), 839-848. · Zbl 0966.58503
[20] W. M. Tulczyjew: The Lagrange complex. Bull. Soc. Math. France 105 (1977), 419-431. · Zbl 0408.58020
[21] A. M. Vinogradov: On the algebro-geometric foundations of Lagrangian field theory. Soviet Math. Dokl. 18 (1977), 1200-1204. · Zbl 0403.58005
[22] A. M. Vinogradov: 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. · Zbl 0406.58015
[23] R. Vitolo: On different geometric formulations of Lagrangian formalism. Diff. Geom. and its Appl. 10 (1999), 225-255. · Zbl 0930.58001
[24] R. Vitolo: Finite order Lagrangian bicomplexes. Math. Proc. Cambridge Phil. Soc. 125 (1998), 321-333. · Zbl 0927.58008
[25] R. Vitolo: A new infinite order formulation of variational sequences. Arch. Math. Univ. Brunensis 34 (1998), 483-504. · Zbl 0970.58002
[26] R. O. Wells: Differential Analysis on Complex Manifolds (GTM, n. 65). Springer-Verlag, Berlin, 1980. · Zbl 0435.32004
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