Let $\pi:Y\rightarrow X$ be a fibered manifold with fibered coordinates $(U,x^i,y^\rho)$ (whenever local calculations are needed) and $m$ dimensional base $X$ ($\dim X=m$).
Suppose that $L\in\Omega^m_{\mathrm{hor}}(J^rY)$ is a Lagrangian of order $r$. As it is well-known (if $L$ is lifted into the variational bicomplex $\Omega^{m,0}(J^\infty Y)$) its vertical differential splits as $$ \delta L=E(L)+\mathrm d_H\theta, $$ where $E(L)$ is the Euler-Lagrange form and $\theta\in\Omega^{m-1,1}$ is an $m-1$-horizontal, $1$-contact form.
It is also well-known that when one works in a fibered chart for $\pi$, the form $\theta$ can be fashioned out of the coefficients of the Lagrangian in that if $L=\mathcal L(x,y,y_{(1)},\dots,y_{(r)})\mathrm d^mx$ then we have $\theta=\theta^i\mathrm\wedge d^{m-1}x_i$ with $$ \theta^i=\sum_{k=0}^{r-1}P^{ii_1...i_k}_\rho\delta y^\rho_{i_1...i_k}, $$where the $P^{ii_1...i_k}_\rho$ are obtained by appropriately differentiating the Lagrangian (too lazy to derive/look-up/remember the exact formulae right now).
Let us call $P^{ii_1...i_k}_\rho$ the order $k+1$ Lagrangian momentum associated to the fibered chart.
What is also known is that the form $\theta$ is not uniquely determined by the Lagrangian, the previous $\theta$ defined in terms of the Lagrangian momenta is not a globally defined differential form and as such not only the individual Lagrangian momenta not transform in any way "tensorially" but even the total combination $\sum_{k=0}^{r-1}P^{ii_1...i_k}\delta y^\rho_{i_1...i_k}\wedge\mathrm d^{m-1}x_i$ is not "tensorial".
It can however be shown that a global $\theta$ exists such that the first variational formula holds, for example by proving the exactness of the horizontal sequences of the variational bicomplex in positive vertical degree or by applying a partition-of-unity argument to the local Lepage forms $\Theta=L-\theta$ (where here $\theta$ is given by the local construction outlined above). This global $\theta$ is however not canonically constructed out of the Lagrangian unless $r=1,2$ or $m=1$.
My problem is that I want to give global and individual existence to the Lagrangian momenta $P^{ii_1...i_k}_\rho$ as the components of some kind of globally defined geometric object. With the momenta associated to a fibered chart this is not possible.
On the other hand let now $\theta\in\Omega^{m-1,1}(J^\infty Y)$ be a global form which splits $\delta$ on the Lagrangian and suppose that in any fibered chart we have the expansion $$ \theta=\sum_{k=0}^{r-1}Q^{ii_1...i_k}_\rho\delta y^\rho_{i_1...i_k}\wedge \mathrm d^{m-1}x_i. $$ The individual $Q^{ii_1...i_k}_\rho$ are still not the components of a global object, however let us take a vertical vector field $\Xi\in\mathfrak X_V(Y)$ on $\pi:Y\rightarrow X$ and hook $\theta$ with its prolongation to get $$ j^\infty\Xi\ \lrcorner\ \theta=\sum_{k=0}^{r-1}Q^{ii_1...i_k}_\rho\Xi^\rho_{i_1...i_k}\mathrm d^{m-1}x_i, $$ where the vertical vector field had the coordinate form $\Xi=\Xi^\rho\frac{\partial}{\partial y^\rho}$ and its prolongation has coordinate form $j^\infty\Xi=\sum_{k=0}^{\infty}\Xi^\rho_{i_1...i_k}\frac{\partial}{\partial y^\rho_{i_1...i_k}}$ and $\Xi^\rho_{i_1...i_k}=D_{i_1}\dots D_{i_k}\Xi^\rho$ (total derivatives).
If an appropriate set of connections are introduced then I am looking to rewrite this as $$ j^\infty\Xi\ \lrcorner\ \theta=\sum_{k=0}^{r-1}\hat Q^{ii_1...i_k}_\rho\nabla_{i_1...i_k}\Xi^\rho\mathrm d^{m-1}x_i, $$ where the $\nabla_{i_1...i_k}=\nabla_{(i_1}\dots\nabla_{i_k)}$ are appropriately symmetrized covariant derivatives and the coefficiens $\hat Q^{ii_1...i_k}_\rho$ are now "tensorial" and can be given individual meaning.
I know that this is possible because I have seen verbal references to constructing global Poincaré-Cartan or Lepage forms via the means of connections and a similar construction is given in the book Fatibene, Francaviglia: Natural and Gauge-Natural formalism for classical field theories however I find the book does not have enough details when these covariant momenta are introduced and there might be errors there as well or at least the treatment is unclear to me.
Some specific points of uncertainty are what connections are precisely required to carry out this procedure. For example since the vertical vector field $\Xi$ is a section of $VY\rightarrow Y$ a linear connection on the vertical tangent bundle should be given, but that alone does not allow to take iterated covariant derivatives, therefore a (probably torsionless) linear connection on $TX\rightarrow X$ also should be given.
However it is still somewhat unclear to me what sort of geometric objects and structures are involved here (for example the coefficients $\nabla_{i_1...i_k}\Xi^\rho$ should probably be interpreted as some coordinate system for $J^{r-1}(VY\rightarrow Y)$ but at this point this is getting confusing).
A specific use case I have in mind is when $Y\rightarrow X$ is a natural fibre bundle, one of the field variables is a metric tensor and all connections are induced from its Levi-Civita connection as in this case the connection is canonically given and the form $\theta$ is also canonically given globally through this construction. However it then means that the Levi-Civita connection $\nabla_{\mathrm{LC}}$ should induce a (in general nonlinear) connection on $\pi:Y\rightarrow X$ (when the order of $\pi$ is $1$, I know how to do this, since then this is just the usual associated bundle construction, but I am not sure how things work when $\pi$ is higher order and is associated to the higher frame bundle $L^s(M)$ for $s\ge 2$), and it should induce a further linear connection on $VY\rightarrow Y$.
For what I want I will need to work with local formulae so it would also be essential to understand how to express both the covariant derivatives $\nabla_{i_1...i_k}\Xi^\rho$ in terms of the Christoffel symbols $\Gamma^{i}_{\ jk}$ of the metric as well as the covariant momenta $\hat Q^{ii_1...i_k}_\rho$ in terms of the connection $\nabla_{\mathrm{LC}}$ and the Lagrangian.
The TL;DR is that I am looking for references which construct the global first variation formula or equivalently global Poincaré-Cartan/Lepage forms in terms of connections such that the "momenta" of the Lagrangian as defined above are in a sense tensorial objects.
I would especially find references valuable which do this in the context of natural fibre bundles and natural variational principles or at least uses that as an application.
