Suppose I have some fibered manifold $E$ with n-dimensional Riemannian base $M$ and sections $\theta:M\rightarrow E$. I have a functional of the form:
$$I=\intop_{M}\mathcal{L}(\theta)d\mu(\theta) $$
Where $\mathcal{L}$ is a lagrangian and a function of $\theta$ and it's derivatives up to rth-order. $d\mu$ is a volume form on $M$. Basically a standard setup for a variational problem.
I'm just learning about Jet bundles, and I'm wondering if it makes any sense at all to take the Jet prolongation of the functional such as:
$$J_{\theta}^{r}I=J_{\theta}^{r}\intop_{M}\mathcal{L}(\theta)d\mu(\theta)=\intop_{M}\mathcal{L}(j_{\theta}^{r}\theta)d\mu(j_{\theta}^{r}\theta)=\intop_{M}\mathcal{L}(\Theta)d\mu(\Theta)$$
Where $$j^{r}\theta=\Theta:M\rightarrow J_{\theta}^{r}E$$ are sections of the $r$-jet bundle or r-jet prolongation of E. I'm coming from a physics background and trying to self-learn jets and prolongations. Ive seen the prolongation of manifolds and sections of bundles but haven't come across anything quite like this yet.
Could someone please explain if this makes any sense or not? My thought was maybe it makes sense if $\mathcal{L}(\Theta)$ are Lepage forms?
To give some motivation to my question, I was thinking that if $\theta$ is a solution to the differential equation defined by finding the stationary points:
$$0=\delta\intop_{M}\mathcal{L}\left(\theta\right)d\mu\left(\theta\right)$$Then $j^{r}\theta=\Theta$ is a solution to the differential equation on the r-jet bundle defined by
$$J^{r}\left(\delta\intop_{M}\mathcal{L}\left(\theta\right)d\mu\left(\theta\right)\right)=0$$ $$=\delta\left(J^{r}\intop_{M}\mathcal{L}\left(\theta\right)d\mu\left(\theta\right)\right)$$ $$=\delta\left(\intop_{M}\mathcal{L}\left(j^{r}\theta\right)d\mu\left(j^{r}\theta\right)\right)$$ $$=\delta\intop_{J^{r}M}\mathcal{L}\left(\mathcal{\Theta}\right)d\mu\left(\Theta\right)$$
Which is based on the Prolongation acting as a coordinate invariant Taylor expansion and thus commutes with the variation $\delta J^{r}=J^{r}\delta$ (? not sure if that's true?). The reason I mention Lepage forms, is I would expect there to be a certain behavior in projecting back to the base space $M$, which is how Lepage forms on jet bundles behave. I'm new to to this, so this question may not even make sense (I apologize if that's the case)