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Frame bundle; Euler-Lagrange equations; invariant Lagrangian; Euler-Poincaré reduction
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.
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