Lagrangians and Euler morphisms on fibered-fibered frame bundles from projectable-projectable classical linear connections

We classify all ${\mathcal{F}}^2{\mathcal{M}}_{m_1,m_2,n_1,n_2}$-natural operators $A$ transforming projectable-projectable torsion-free classical linear connections $\mathrm{\nabla }$ on fibered-fibered manifolds $Y$ of dimension $(m_1,m_2,n_1,n_2)$ into $r$th order Lagrangians $A(r)$ on the fibered-fibered linear frame bundle $L^{fib-fib}(Y)$ on $Y$. Moreover, we classify all ${\mathcal{F}}^2{\mathcal{M}}_{m_1,m_2,n_1,n_2}$-natural operators $B$ transforming projectable-projectable torsion-free classical linear connections r on fiberedfibered manifolds $Y$ of dimension $(m_1,m_2,n_1,n_2)$ into Euler morphism $B(\mathrm{\nabla })$ on $L^{fib-fib}(Y)$. These classifications can be expanded on the $k$th order fibered-fibered frame bundle $L^{fib-fib,k}(Y)$ instead of $L^{fib-fib}(Y)$.