Geodesic
Order reduction of the Euler-Lagrange equations of higher order invariant variational problems on frame bundles
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 1063-1076

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Let $\mu \colon FX \to X$ 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.
DOI : 10.1007/s10587-011-0048-4
Classification : 53C05, 53C10, 58A20, 58E30
Keywords: Frame bundle; Euler-Lagrange equations; invariant Lagrangian; Euler-Poincaré reduction 
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     title = {Order reduction of the {Euler-Lagrange} equations of higher order invariant variational problems on frame bundles},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1063--1076},
     publisher = {mathdoc},
     volume = {61},
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     year = {2011},
     doi = {10.1007/s10587-011-0048-4},
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     zbl = {1249.53029},
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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, Tome 61 (2011) no. 4, pp. 1063-1076. doi: 10.1007/s10587-011-0048-4

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