Invariant variational problems on principal bundles and conservation laws

Ján Brajerčík

Archivum Mathematicum (2011)

  • Volume: 047, Issue: 5, page 357-366
  • ISSN: 0044-8753

Abstract

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In this work, we consider variational problems defined by G -invariant Lagrangians on the r -jet prolongation of a principal bundle P , where G is the structure group of P . These problems can be also considered as defined on the associated bundle of the r -th order connections. The correspondence between the Euler-Lagrange equations for these variational problems and conservation laws is discussed.

How to cite

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Brajerčík, Ján. "Invariant variational problems on principal bundles and conservation laws." Archivum Mathematicum 047.5 (2011): 357-366. <http://eudml.org/doc/246116>.

@article{Brajerčík2011,
abstract = {In this work, we consider variational problems defined by $G$-invariant Lagrangians on the $r$-jet prolongation of a principal bundle $P$, where $G$ is the structure group of $P$. These problems can be also considered as defined on the associated bundle of the $r$-th order connections. The correspondence between the Euler-Lagrange equations for these variational problems and conservation laws is discussed.},
author = {Brajerčík, Ján},
journal = {Archivum Mathematicum},
keywords = {principal bundle; variational principle; invariant Lagrangian; Euler-Lagrange equations; Noether’s current; conservation law; principal bundle; variational principle; invariant Lagrangian; Euler-Lagrange equations; Noether's current; conservation law},
language = {eng},
number = {5},
pages = {357-366},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Invariant variational problems on principal bundles and conservation laws},
url = {http://eudml.org/doc/246116},
volume = {047},
year = {2011},
}

TY - JOUR
AU - Brajerčík, Ján
TI - Invariant variational problems on principal bundles and conservation laws
JO - Archivum Mathematicum
PY - 2011
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 047
IS - 5
SP - 357
EP - 366
AB - In this work, we consider variational problems defined by $G$-invariant Lagrangians on the $r$-jet prolongation of a principal bundle $P$, where $G$ is the structure group of $P$. These problems can be also considered as defined on the associated bundle of the $r$-th order connections. The correspondence between the Euler-Lagrange equations for these variational problems and conservation laws is discussed.
LA - eng
KW - principal bundle; variational principle; invariant Lagrangian; Euler-Lagrange equations; Noether’s current; conservation law; principal bundle; variational principle; invariant Lagrangian; Euler-Lagrange equations; Noether's current; conservation law
UR - http://eudml.org/doc/246116
ER -

References

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  6. Castrillón López, M., Ratiu, T. S., Shkoller, S., 10.1090/S0002-9939-99-05304-6, Proc. Amer. Math. Soc. 128 (7) (2000), 2155–2164. (2000) MR1662269DOI10.1090/S0002-9939-99-05304-6
  7. Kobayashi, S., Nomizu, K., Foundations of Differential Geometry, Vol. 1, 2, Interscience Publishers, Wiley, New York, 1963. (1963) MR0152974
  8. Kolář, I., Michor, P. W., Slovák, J., Natural Operations in Differential Geometry, Springer Verlag, Berlin, 1993. (1993) MR1202431
  9. Krupka, D., Some geometric aspects of variational problems in fibered manifold, Folia Fac. Sci. Natur. Univ. Purk. Brun. Phys. 14 (1973). (1973) 
  10. Krupka, D., 10.1016/0022-247X(75)90190-0, J. Math. Anal. Appl. 49 (1975), 469–476. (1975) Zbl0312.58003MR0362398DOI10.1016/0022-247X(75)90190-0
  11. Krupka, D., Lepagean forms in higher order variational theory, Proc. IUTAM-ISIMM Symposium, Modern Developements in Analytical Mechanics I: Geometrical Dynamics (Benenti, S., Francaviglia, M., Lichnerowicz, A., eds.), Accad. delle Scienze di Torino, Torino, 1983, pp. 197–238. (1983) Zbl0572.58003MR0773488
  12. Krupka, D., Janyška, J., Lectures on Differential Invariants, Folia Fac. Sci. Natur. Univ. Purk. Brun. Math. 1 (1990). (1990) MR1108622
  13. Trautman, A., Invariance of Lagrangian systems, General RelativityPapers in honour of J. L. Synge, pp. 85–99, Clarendon Press, Oxford, 1972. (1972) MR0503424

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