I'm working to simplify the Lagrangian formalism of classical field theory for the situation of a vector bundle with a bundle metric and a metric connection. Particularly, I want to specify the Euler-Lagrange equations and the Noether theorem for this case.
Consider a vector bundle $(E,\pi,M,\mathbb R^n)$ with a bundle metric $g$ and a metric connection $\nabla$. Let $J^1E$ be the 1-jet bundle associated to $E$. Is there a canonical way to identify an element $j \in J^1E$ with an element $(\phi,\nabla \phi) \in E\times(E\otimes TM^*)$? I would also be grateful for some bibliography on that subject.
What I know is that there exists a 1:1 correspondence between sections of $J^1E \to E$ and connections on $E$. Furthermore the connection leads uniquely to a splitting $TE = VE \oplus HE$ of the tangent bundle of $E$.