Order reduction of the Euler-Lagrange equations of higher order invariant variational problems on frame bundles

Ján Brajerčík

Czechoslovak Mathematical Journal (2011)

  • Volume: 61, Issue: 4, page 1063-1076
  • ISSN: 0011-4642

Abstract

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Let μ : F X → X be a principal bundle of frames with the structure group Gl n ( ℝ ) . It is shown that the variational problem, defined by Gl n ( ℝ ) -invariant Lagrangian on J r F X , can be equivalently studied on the associated space of connections with some compatibility condition, which gives us order reduction of the corresponding Euler-Lagrange equations.

How to cite

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Brajerčík, Ján. "Order reduction of the Euler-Lagrange equations of higher order invariant variational problems on frame bundles." Czechoslovak Mathematical Journal 61.4 (2011): 1063-1076. <http://eudml.org/doc/196784>.

@article{Brajerčík2011,
abstract = {Let $\mu \colon FX \rightarrow X$ be a principal bundle of frames with the structure group $\{\rm Gl\}_\{n\}(\mathbb \{R\})$. It is shown that the variational problem, defined by $\{\rm Gl\}_\{n\}(\mathbb \{R\})$-invariant Lagrangian on $J^\{r\} FX$, can be equivalently studied on the associated space of connections with some compatibility condition, which gives us order reduction of the corresponding Euler-Lagrange equations.},
author = {Brajerčík, Ján},
journal = {Czechoslovak Mathematical Journal},
keywords = {Frame bundle; Euler-Lagrange equations; invariant Lagrangian; Euler-Poincaré reduction; frame bundle; Euler-Lagrange equations; invariant Lagrangian; Euler-Poincaré reduction},
language = {eng},
number = {4},
pages = {1063-1076},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Order reduction of the Euler-Lagrange equations of higher order invariant variational problems on frame bundles},
url = {http://eudml.org/doc/196784},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Brajerčík, Ján
TI - Order reduction of the Euler-Lagrange equations of higher order invariant variational problems on frame bundles
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 4
SP - 1063
EP - 1076
AB - Let $\mu \colon FX \rightarrow X$ be a principal bundle of frames with the structure group ${\rm Gl}_{n}(\mathbb {R})$. It is shown that the variational problem, defined by ${\rm Gl}_{n}(\mathbb {R})$-invariant Lagrangian on $J^{r} FX$, can be equivalently studied on the associated space of connections with some compatibility condition, which gives us order reduction of the corresponding Euler-Lagrange equations.
LA - eng
KW - Frame bundle; Euler-Lagrange equations; invariant Lagrangian; Euler-Poincaré reduction; frame bundle; Euler-Lagrange equations; invariant Lagrangian; Euler-Poincaré reduction
UR - http://eudml.org/doc/196784
ER -

References

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  1. Brajerčík, J., G l n ( ℝ ) -invariant variational principles on frame bundles, Balkan J. Geom. Appl. 13 (2008), 11-19. (2008) MR2395370
  2. Brajerčík, J., Krupka, D., 10.1063/1.1901323, J. Math. Phys. 46 (2005), 1-15. (2005) MR2143003DOI10.1063/1.1901323
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  6. Dieudonné, J., Treatise on Analysis. Vol. I, Academic Press New York-London (1969). (1969) 
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  12. Krupka, D., Local invariants of a linear connection, In: Differential Geometry. Colloq. Math. Soc. János Bolyai, Budapest, 1979, Vol. 31 North Holland Amsterdam (1982), 349-369. (1982) Zbl0513.53038MR0706930
  13. Krupka, D., Janyška, J., Lectures on Differential Invariants. Folia Fac. Sci. Nat. Univ. Purk. Brunensis, Mathematica 1, University J. E. Purkyně Brno (1990). (1990) MR1108622
  14. Marsden, J. E., Ratiu, T. S., 10.1007/978-1-4612-2682-6, Springer New York (1994). (1994) MR1304682DOI10.1007/978-1-4612-2682-6
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