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Keywords:
jets; gauge-natural bundles; variational principles; generalized Bianchi identities; Jacobi morphisms; invariance and symmetry properties
jets; gauge-natural bundles; variational principles; generalized Bianchi identities; Jacobi morphisms; invariance and symmetry properties
Summary:
We derive both local and global generalized Bianchi identities for classical Lagrangian field theories on gauge-natural bundles. We show that globally defined generalized Bianchi identities can be found without the a priori introduction of a connection. The proof is based on a global decomposition of the variational Lie derivative of the generalized Euler-Lagrange morphism and the representation of the corresponding generalized Jacobi morphism on gauge-natural bundles. In particular, we show that within a gauge-natural invariant Lagrangian variational principle, the gauge-natural lift of infinitesimal principal automorphism is not intrinsically arbitrary. As a consequence the existence of canonical global superpotentials for gauge-natural Noether conserved currents is proved without resorting to additional structures.
References:
[1] Alonso R. J.: Decomposition of higher order tangent fields and calculus of variations. Proc. Diff. Geom. Appl. (Brno, 1998), 451–460, Masaryk Univ., Brno, 1999. MR 1708934
[2] Alonso R. J.: $D$-modules, contact valued calculus and Poincaré-Cartan form. Czechoslovak Math. J. 49 (124) (3) (1999), 585–606. MR 1708350 | Zbl 1011.58011
[3] Anderson J. L., Bergmann P. G.: Constraints in covariant field theories. Phys. Rev. 83 (5) (1951), 1018–1025. MR 0044382 | Zbl 0045.45505
[4] Allemandi G., Fatibene L., Ferraris M., Francaviglia M., Raiteri M.: Nöther conserved quantities and entropy in general relativity. In: Recent Developments in General Relativity, Genoa 2000; R. Cianci et al. eds., Springer Italia, Milano (2001), 75–92. MR 1852664 | Zbl 1202.83039
[5] Bergmann P. G.: Non-linear field theories. Phys. Rev. 75 (4) (1949), 680–685. MR 0028128 | Zbl 0039.23004
[6] Bergmann P. G.: Conservation laws in general relativity as the generators of coordinate transformations. Phys. Rev. 112 (1) (1958), 287–289. MR 0099236
[7] Chruściel P. T.: On the relation between the Einstein and the Komar expressions for the energy of the gravitational field. Ann. Inst. H. Poincaré 42 (3) (1985), 267–282. MR 0797276 | Zbl 0645.53063
[8] Eck D. J.: Gauge-natural bundles and generalized gauge theories. Mem. Amer. Math. Soc. 247 (1981), 1–48. MR 0632164 | Zbl 0493.53052
[9] Fatibene L., Francaviglia M., Raiteri M.: Gauge natural field theories and applications to conservation laws. Proc. VIII Conf. Differential Geom. Appl., O. Kowalski et al. eds.; Silesian University at Opava, Opava (Czech Republic) 2001, 401–413. MR 1978794 | Zbl 1026.70027
[10] Fatibene L., Francaviglia M., Palese M.: Conservation laws and variational sequences in gauge-natural theories. Math. Proc. Cambridge Philos. Soc. 130 (2001), 555–569. MR 1816809 | Zbl 0988.58006
[11] Ferraris M., Francaviglia M.: The Lagrangian approach to conserved quantities in general relativity. In: Mechanics, Analysis and Geometry: 200 Years after Lagrange; M. Francaviglia ed.; Elsevier Science Publishers B. V. (Amsterdam, 1991), 451–488. MR 1098527 | Zbl 0717.53060
[12] Ferraris M., Francaviglia M., Raiteri M.: Conserved quantities from the equations of motion (with applications to natural and gauge natural theories of gravitation). Classical Quantum Gravity 20 (2003), 4043–4066. MR 2017333
[13] Francaviglia M., Palese M.: Second order variations in variational sequences. Steps in Differential Geometry (Debrecen, 2000) Inst. Math. Inform. Debrecen, Hungary (2001), 119–130. MR 1859293 | Zbl 0977.58019
[14] Francaviglia M., Palese M.: Generalized Jacobi morphisms in variational sequences. In: Proc. XXI Winter School Geometry and Physics, Srní 2001, Rend. Circ. Mat. Palermo (2) Suppl. 69 (2002), 195–208. MR 1972435 | Zbl 1028.58022
[15] Francaviglia M., Palese M., Vitolo R.: Symmetries in finite order variational sequences. Czechoslovak Math. J. 52 (127) (2002), 197–213. MR 1885465 | Zbl 1006.58014
[16] Francaviglia M., Palese M., Vitolo R.: Superpotentials in variational sequences. Proc. VII Conf. Differential Geom. Appl., Satellite Conf. of ICM in Berlin (Brno 1998); I. Kolář et al. eds.; Masaryk University in Brno (Czech Republic) 1999, 469–480. MR 1708936
[17] Francaviglia M., Palese M., Vitolo R.: The Hessian and Jacobi morphisms for higher order calculus of variations. Differential Geom. Appl. 22 (1) (2005), 105–120. MR 2106379 | Zbl 1065.58010
[18] Godina M., Matteucci P.: Reductive $G$-structures and Lie derivatives. J. Geom. Phys. 47 (1) (2003), 66–86. MR 1985484 | Zbl 1035.53035
[19] Goldberg J. N.: Conservation laws in general relativity. Phys. Rev. (2) 111 (1958), 315–320. MR 0099235 | Zbl 0089.20903
[20] Goldschmidt H., Sternberg S.: The Hamilton-Cartan formalism in the calculus of variations. Ann. Inst. Fourier, Grenoble 23 (1) (1973), 203–267. MR 0341531 | Zbl 0243.49011
[21] Horák M., Kolář I.: On the higher order Poincaré-Cartan forms. Czechoslovak Math. J. 33 (108) (1983), 467–475. Zbl 0545.58004
[22] Janyška J.: Natural and gauge-natural operators on the space of linear connections on a vector bundle. Proc. Differential Geom. Appl. (Brno, 1989); J Janyška, D. Krupka eds.; World Scientific (Singapore, 1990), 58–68. MR 1062006
[23] Janyška J.: Reduction theorems for general linear connections. Differential Geom. Appl. 20 (2004), no. 2, 177–196. MR 2038554
[24] Janyška J., Modugno M.: Infinitesimal natural and gauge-natural lifts. Differential Geom. Appl. 2 (2) (1992), 99–121. MR 1245551 | Zbl 0780.53023
[25] Julia B., Silva S.: Currents and superpotentials in classical gauge theories, II, Global Aspects and the example of affine gravity. Classical Quantum Gravity 17 (22) (2000), 4733–4743. MR 1797968 | Zbl 0988.83026
[26] Katz J.: A note on Komar’s anomalous factor. Classical Quantum Gravity 2 (3) (1985), 423–425. MR 0792031
[27] Kolář I.: On some operations with connections. Math. Nachr. 69 (1975), 297–306. MR 0391157
[28] Kolář I.: Prolongations of generalized connections. Coll. Math. Soc. János Bolyai, (Differential Geometry, Budapest, 1979) 31 (1979), 317–325. MR 0706928
[29] Kolář I.: A geometrical version of the higher order Hamilton formalism in fibred manifolds. J. Geom. Phys. 1 (2) (1984), 127–137. MR 0794983
[30] Kolář I.: Some geometric aspects of the higher order variational calculus. Geom. Meth. in Phys., Proc. Diff. Geom. and its Appl., (Nové Město na Moravě, 1983); D. Krupka ed.; J. E. Purkyně University (Brno, 1984), 155–166. MR 0793206
[31] Kolář I.: Natural operators related with the variational calculus. Proc. Differential Geom. Appl. (Opava, 1992), 461–472, Math. Publ. 1 Silesian Univ. Opava, Opava, 1993. MR 1255562
[32] Kolář I., Michor P. W., Slovák J.: Natural operations in differential geometry. Springer-Verlag, N.Y., 1993. MR 1202431 | Zbl 0782.53013
[33] Kolář I., Virsik G.: Connections in first principal prolongations. In: Proc. XVI Winter School Geometry and Physics, Srní 1995, Rend. Circ. Mat. Palermo (2), Suppl. 43 (1995), 163–171. MR 1463518
[34] Kolář I., Vitolo R.: On the Helmholtz operator for Euler morphisms. Math. Proc. Cambridge Philos. Soc. 135 (2) (2003), 277–290. MR 2006065 | Zbl 1048.58012
[35] Komar A.: Covariant conservation laws in general relativity. Phys. Rev. 113 (3) (1959), 934–936. MR 0102403 | Zbl 0086.22103
[36] Krupka D.: Variational sequences on finite order jet spaces. Proc. Diff. Geom. and its Appl. (Brno, 1989), J. Janyška, D. Krupka eds.; World Scientific (Singapore, 1990), 236–254. MR 1062026
[37] Krupka D.: Topics in the calculus of variations: finite order variational sequences. O. Kowalski and D. Krupka eds., Proc. Differential Geom. and its Appl. (Opava, 1992), Math. Publ. 1, Silesian Univ. Opava, Opava, 1993, 473–495. MR 1255563
[38] Mangiarotti L., Modugno M.: Fibered spaces, jet spaces and connections for field theories. In: Proc. Int. Meet. Geom. Phys.; M. Modugno ed.; Pitagora Editrice (Bologna, 1983), 135–165. MR 0760841 | Zbl 0539.53026
[39] Nöther E.: Invariante variationsprobleme. Nachr. Ges. Wiss. Gött., Math. Phys. Kl. II (1918), 235–257.
[40] Palese M.: Geometric foundations of the calculus of variations. Variational sequences, symmetries and Jacobi morphisms. Ph.D. Thesis, University of Torino (2000).
[41] Palese M., Winterroth E.: In preparation. Zbl 1233.58002
[42] Saunders D. J.: The geometry of jet bundles. Cambridge Univ. Press (Cambridge, 1989). MR 0989588 | Zbl 0665.58002
[43] Trautman A.: Conservation laws in general relativity. In Gravitation: An introduction to current research, pp. 169–198, Wiley, New York 1962. MR 0143627
[44] Trautman A.: Noether equations and conservation laws. Comm. Math. Phys. 6 (1967), 248–261. MR 0220470 | Zbl 0172.27803
[45] Trautman A.: A metaphysical remark on variational principles. Acta Phys. Polon. B XX (1996), 1–9. MR 1388335 | Zbl 0966.58503
[46] Vitolo R.: Finite order Lagrangian bicomplexes. Math. Proc. Camb. Phil. Soc. 125 (1) (1999), 321–333. MR 1643802
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