Geodesic
The relation between the Jacobi morphism and the Hessian in gauge-natural field theories
Teoretičeskaâ i matematičeskaâ fizika, Tome 152 (2007) no. 2, pp. 377-389 Cet article a éte moissonné depuis la source Math-Net.Ru

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We generalize a classic result, due to Goldschmidt and Sternberg, relating the Jacobi morphism and the Hessian for first-order field theories to higher-order gauge-natural field theories. In particular, we define a generalized gauge-natural Jacobi morphism where the variation vector fields are Lie derivatives of sections of the gauge-natural bundle with respect to gauge-natural lifts of infinitesimal principal automorphisms, and we relate it to the Hessian. The Hessian is also very simply related to the generalized Bergmann–Bianchi morphism, whose kernel provides necessary and sufficient conditions for the existence of global canonical superpotentials. We find that the Hamilton equations for the Hamiltonian connection associated with a suitably defined covariant strongly conserved current are satisfied identically and can be interpreted as generalized Bergmann–Bianchi identities and thus characterized in terms of the Hessian vanishing.
Mots-clés : jet
Keywords: gauge-natural bundle, second variational derivative, generalized Jacobi morphism. 
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M. Palese; E. Winterroth. The relation between the Jacobi morphism and the Hessian in gauge-natural field theories. Teoretičeskaâ i matematičeskaâ fizika, Tome 152 (2007) no. 2, pp. 377-389. http://geodesic.mathdoc.fr/item/TMF_2007_152_2_a12/

[1] H. Goldschmidt, S. Sternberg, Ann. Inst. Fourier, 23:1 (1973), 203–267 | DOI | MR | Zbl

[2] A. V. Bocharov, A. M. Verbovetskii, A. M. Vinogradovi dr., Simmetrii i zakony sokhraneniya uravnenii matematicheskoi fiziki, Faktorial, M., 1997 | MR | Zbl

[3] I. Kolář, P. W. Michor, J. Slovák, Natural Operations in Differential Geometry, Springer, Berlin, 1993 | MR | Zbl

[4] D. Krupka, “Some geometric aspects of variational problems in fibred manifolds”, Folia Fac. Sci. Nat. Univ. Purk. Brunensis, Physica, 14, J. E. Purkyně Univ., Brno, 1973, 1–65 ; math-ph/0110005

[5] B. A. Kuperschmidt, “Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalism”, Geometric Methods in Mathematical Physics, Proc. NSF-CBMS Conf. (Lowell, 1979), Lect. Notes Math., 775, eds. G. Kaiser, J. E. Marsden, Springer, Berlin, 1980, 162–218 | DOI | MR | Zbl

[6] P. J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed., Grad. Texts in Math., 107, Springer, New York, 1993 | DOI | MR | Zbl

[7] R. S. Palais, Foundations of Global Non-linear Analysis, Benjamin, New York–Amsterdam, 1968 | MR | Zbl

[8] D. J. Saunders, The Geometry of Jet Bundles, London Math. Soc. Lect. Note Ser., 142, Cambridge Univ. Press, Cambridge, 1989 | MR | Zbl

[9] A. M. Vinogradov, J. Math. Anal. Appl., 100:1 (1984), 1–40 ; 41–129 | DOI | MR | Zbl | DOI | MR | Zbl

[10] F. Takens, J. Diff. Geom., 14 (1979), 543–562 | DOI | MR | Zbl

[11] W. M. Tulczyjew, Bull. Soc. Math. France, 105 (1977), 419–431 | DOI | MR | Zbl

[12] A. M. Vinogradov, DAN SSSR, 236:2 (1977), 284–287 | MR | Zbl

[13] A. M. Vinogradov, DAN SSSR, 238:5 (1978), 1028–1031 | MR | Zbl

[14] R. Vitolo, Diff. Geom. Appl., 10:3 (1999), 225–255 | DOI | MR | Zbl

[15] P. G. Bergmann, Phys. Rev., 75 (1949), 680–685 | DOI | MR | Zbl

[16] E. Neter, “Invariantnye variatsionnye zadachi”, Variatsionnye printsipy mekhaniki, Fizmatlit, M., 1959, 604–630 | MR | Zbl

[17] L. Fatibene, M. Francaviglia, M. Palese, Math. Proc. Cambridge Philos. Soc., 130 (2001), 555–569 | DOI | MR | Zbl

[18] P. Matteucci, Rep. Math. Phys., 52:1 (2003), 115–139 | DOI | MR | Zbl

[19] J. L. Anderson, P. G. Bergmann, Phys. Rev., 83:5 (1951), 1018–1025 | DOI | MR | Zbl

[20] M. Palese, E. Winterroth, Arch. Math. (Brno), 41:3 (2005), 289–310 | MR | Zbl

[21] M. Palese, E. Winterroth, Rep. Math. Phys., 54:3 (2004), 349–364 | DOI | MR | Zbl

[22] M. Francaviglia, M. Palese, E. Winterroth, Rep. Math. Phys., 56:1 (2005), 11–22 ; math-ph/0407054 | DOI | MR | Zbl

[23] M. Francaviglia, M. Palese, E. Winterroth, “Second variational derivative of gauge-natural invariant Lagrangians and conservation laws”, Differential Geometry and its Applications, Proc. IX Int. Conf. (Prague, Czech Republic, 2004), eds. J. Bures et al., Matfyzpress, Prague, 2005, 591–604 | MR | Zbl

[24] M. Palese, E. Winterroth, “Gauge-natural field theories and Noether theorems: canonical covariant conserved currents”, Geometry and Physics (Srni, CZ, 2005), Rend. Circ. Mat., Palermo, 2006, 161–174 | MR | Zbl

[25] M. Francaviglia, M. Palese, R. Vitolo, Diff. Geom. Appl., 22:1 (2005), 105–120 | DOI | MR | Zbl

[26] M. Giaquinta, S. Hildebrandt, Calculus of Variations. Vol. I. The Lagrangian Formalism, Grundlehren Math. Wiss., 310, Springer, Berlin, 1996 | MR | Zbl

[27] R. Vitolo, Math. Proc. Cambridge Philos. Soc., 125:1 (1999), 321–333 | DOI | MR | Zbl

[28] D. J. Eck, Mem. Amer. Math. Soc., 33:247 (1981), 1–48 | MR | Zbl

[29] I. Kolář, J. Geom. Phys., 1:2 (1984), 127–137 | DOI | MR | Zbl

[30] I. Kolář, R. Vitolo, Math. Proc. Cambridge Philos. Soc., 135:2 (2003), 277–290 | DOI | MR | Zbl

[31] D. Krupka, “Variational sequences on finite order jet spaces”, Differential Geometry and its Applications (Brno, Czechoslovakia, 1989), eds. J. Janyška, D. Krupka, World Scientific, Singapore, 1990, 236–254 | MR | Zbl

[32] M. Francaviglia, M. Palese, R. Vitolo, Czechoslovak Math. J., 52(127):1 (2002), 197–213 | DOI | MR | Zbl

[33] M. Godina, P. Matteucci, Int. J. Geom. Meth. Mod. Phys., 2 (2005), 159–188 | DOI | MR

[34] A. Trautman, Commun. Math. Phys., 6 (1967), 248–261 | DOI | MR | Zbl

[35] L. Mangiarotti, G. Sardanashvily, Connections in Classical and Quantum Field Theory, World Scientific, Singapore, 2000 | MR | Zbl