Let $\mu :FX\to X$ be a principal bundle of frames with the structure group ${\mathrm{Gl}}_n\left(ℝ\right)$. It is shown that the variational problem, defined by ${\mathrm{Gl}}_n\left(ℝ\right)$-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.