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Context: I am a PhD student in theoretical physics with higher-than-average education on differential geometry. I am trying to understand Lagrangian and Hamiltonian field theories and related concepts ...

In what follows, all manifolds are smooth, Hausdorff, paracompact, connected and oriented, and all maps between any two of them are assumed to be smooth. Let $\pi:E\rightarrow M$ be a fiber bundle ...

Let $\pi:N\rightarrow M$ be a fibered manifold with $m=\dim M$ and $m+n=\dim N$. The variational bicomplex on the infinite jet space $J^\infty(\pi)$ is denoted $(\Omega^{k,l}(\pi),\delta,\mathbf d)$ ...