| Title:
|
Second variational derivative of local variational problems and conservation laws (English) |
| Author:
|
Palese, Marcella |
| Author:
|
Winterroth, Ekkehart |
| Author:
|
Garrone, E. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
47 |
| Issue:
|
5 |
| Year:
|
2011 |
| Pages:
|
395-403 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider cohomology defined by a system of local Lagrangian and investigate under which conditions the variational Lie derivative of associated local currents is a system of conserved currents. The answer to such a question involves Jacobi equations for the local system. Furthermore, we recall that it was shown by Krupka et al. that the invariance of a closed Helmholtz form of a dynamical form is equivalent with local variationality of the Lie derivative of the dynamical form; we remark that the corresponding local system of Euler–Lagrange forms is variationally equivalent to a global one. (English) |
| Keyword:
|
fibered manifold |
| Keyword:
|
jet space |
| Keyword:
|
Lagrangian formalism |
| Keyword:
|
variational sequence |
| Keyword:
|
second variational derivative |
| Keyword:
|
cohomology |
| Keyword:
|
symmetry |
| Keyword:
|
conservation law |
| MSC:
|
55N30 |
| MSC:
|
55R10 |
| MSC:
|
58A12 |
| MSC:
|
58A20 |
| MSC:
|
58E30 |
| MSC:
|
70S10 |
| idZBL:
|
Zbl 1265.58008 |
| idMR:
|
MR2876943 |
| . |
| Date available:
|
2011-12-16T15:28:23Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141787 |
| . |
| Reference:
|
[1] Allemandi, G., Francaviglia, M., Raiteri, M.: Covariant charges in Chern-Simons $AdS3$ gravity.Classical Quantum Gravity 20 (3) (2003), 483–506. MR 1957170, 10.1088/0264-9381/20/3/307 |
| Reference:
|
[2] Anderson, I. M., Duchamp, T.: On the existence of global variational principles.Amer. Math. J. 102 (1980), 781–868. Zbl 0454.58021, MR 0590637, 10.2307/2374195 |
| Reference:
|
[3] Borowiec, A., Ferraris, M., Francaviglia, M., Palese, M.: Conservation laws for non-global Lagrangians.Univ. Iagel. Acta Math. 41 (2003), 319–331. Zbl 1060.70034, MR 2084774 |
| Reference:
|
[4] Brajerčík, J., Krupka, D.: Variational principles for locally variational forms.J. Math. Phys. 46 (5) (2005), 15, 052903. Zbl 1110.58011, MR 2143003, 10.1063/1.1901323 |
| Reference:
|
[5] Dedecker, P., Tulczyjew, W. M.: Spectral sequences and the inverse problem of the calculus of variations.Lecture Notes in Math., vol. 836, Springer–Verlag, 1980, pp. 498–503. Zbl 0482.49027, MR 0607719 |
| Reference:
|
[6] Eck, D. J.: Gauge-natural bundles and generalized gauge theories.Mem. Amer. Math. Soc. 247 (1981), 1–48. Zbl 0493.53052, MR 0632164 |
| Reference:
|
[7] 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, 10.1088/0264-9381/20/18/312 |
| Reference:
|
[8] Ferraris, M., Palese, M., Winterroth, E.: Local variational problems and conservation laws.Differential Geom. Appl. 29 (2011), S80–S85. Zbl 1233.58002, MR 2832003, 10.1016/j.difgeo.2011.04.011 |
| Reference:
|
[9] Francaviglia, M., Palese, M.: Second order variations in variational sequences.Steps in differential geometry. Proceedings of the colloquium on differential geometry (Kozma, L. et al., ed.), Univ. Debrecen, Institute of Mathematics and Informatics, 2001, Debrecen, Hungary, July 25-30, 2000, pp. 119–130. Zbl 0977.58019, MR 1859293 |
| Reference:
|
[10] Francaviglia, M., Palese, M., Vitolo, R.: Symmetries in finite order variational sequences.Czechoslovak Math. J. 52 (127) (1) (2002), 197–213. Zbl 1006.58014, MR 1885465, 10.1023/A:1021735824163 |
| Reference:
|
[11] 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. Zbl 1065.58010, MR 2106379, 10.1016/j.difgeo.2004.07.008 |
| Reference:
|
[12] Krupka, D.: Some Geometric Aspects of Variational Problems in Fibred Manifolds.Folia Fac. Sci. Natur. UJEP Brunensis, vol. 14, 1973, pp. 1–65. |
| Reference:
|
[13] Krupka, D.: Variational Sequences on Finite Order Jet Spaces.Proc. Differential Geom. Appl. (Janyška, J., Krupka, D., eds.), World Sci. Singapore, 1990, pp. 236–254. Zbl 0813.58014, MR 1062026 |
| Reference:
|
[14] Krupka, D., Krupková, O., Prince, G., Sarlet, W.: Contact symmetries of the Helmholtz form.Differential Geom. Appl. 25 (5) (2007), 518–542. MR 2351428, 10.1016/j.difgeo.2007.06.003 |
| Reference:
|
[15] Palese, M., Winterroth, E.: Covariant gauge-natural conservation laws.Rep. Math. Phys. 54 (3) (2004), 349–364. Zbl 1066.58009, MR 2115744, 10.1016/S0034-4877(04)80024-7 |
| Reference:
|
[16] Palese, M., Winterroth, E.: Global generalized Bianchi identities for invariant variational problems on Gauge-natural bundles.Arch. Math. (Brno) 41 (3) (2005), 289–310. Zbl 1112.58005, MR 2188385 |
| Reference:
|
[17] Palese, M., Winterroth, E.: Variational Lie derivative and cohomology classes.AIP Conf. Proc. 1360 (2011), 106–112. |
| Reference:
|
[18] Sardanashvily, G.: Noether conservation laws issue from the gauge invariance of an Euler-Lagrange operator, but not a Lagrangian.arXiv:math-ph/0302012; see also Bashkirov, D., Giachetta, G., Mangiarotti, L., Sardanashvily, G.: Noether's second theorem in a general setting. Reducible gauge theories, J. Phys. A38 (2005), 5329–5344. |
| Reference:
|
[19] Takens, F.: A global version of the inverse problem of the calculus of variations.J. Differential Geom. 14 (1979), 543–562. Zbl 0463.58015, MR 0600611 |
| Reference:
|
[20] Tulczyjew, W. M.: The Lagrange complex.Bull. Soc. Math. France 105 (1977), 419–431. Zbl 0408.58020, MR 0494272 |
| Reference:
|
[21] Vinogradov, A. M.: On the algebro–geometric foundations of Lagrangian field theory.Soviet Math. Dokl. 18 (1977), 1200–1204. Zbl 0403.58005, MR 0501142 |
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