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I am learning classical field theory at the moment. Right now, I am trying to understand where the Legendre transform, or the multi-symplectic structure emerges.

To make the setting clear, say we have a fiber bundle $Y \overset{\pi}{\to} X$ over a $n$-manifold $X$. A Lagrangian density is a $n$-form on $X$ depending on the $1$-jet of a section of $Y$ : $\mathcal{L}\in\Gamma(J^1\pi,\pi^*\Lambda^n X)$.

The affine Legendre transform is a map $J^1\pi \to \Lambda^n_1 Y$ that is essentially the affine approximation to $\mathcal L$ at each point:

$$ FL : s \mapsto \mathcal{L}(s) + d_V\mathcal{L} \in \Lambda^n X \oplus \Lambda^n X\otimes V^* \simeq \Lambda^n_1 Y $$

where $V$ is the vertical bundle of $\pi$ and $d_V$ the vertical derivative, well-defined because $\pi^*\Lambda^n X$ is trivial above fibers of $\pi$.

In this approach, the Poincaré-Cartan $n$-form is defined by $\Theta_{\mathcal L} = FL^*\Theta$ where $\Theta$ is the tautological form on $\Lambda^n T^*Y$.

The key feature of the Poincaré-Cartan form is that $\mathcal L - \Theta_{\mathcal L}$ is contact, that is to say it vanishes on holonomic sections of $J^1Y$, that is $1$-jet of sections of $Y$. Hence there is the fundamental identity \begin{equation}\tag{1}\label{contact} \forall \phi\in\Gamma(\pi), j^1(\phi)^*\Theta_{\mathcal L} = j^1(\phi)^*\mathcal L = {\mathcal L}\circ j^1(\phi) \end{equation}

Now, these forms can be obtained from a variational principle [1]: for $\phi$ a section over $U_X$, the action $S(\phi)$ is defined as $\int_{U_X} \mathcal{L}\circ j^1(\phi)$. For a variation $V$ of $\phi$, the variation of $S$ can be expressed as: $$ dS_\phi\cdot V = \int_{U_X} j^1(\phi)^*(j^1(V)\lrcorner d\Theta_{\mathcal L}) + \int_{\partial U_X} j^1(\phi)^*(j^1(V)\lrcorner \Theta_{\mathcal L}) $$

Now, the article [1] mentions at p.27 (Corollary 4.2) that from this very identity we can obtain (\ref{contact}). I am missing an argument, so I am unable to derive (\ref{contact}), but I am very interested by the perspective of justifying the Legendre transform from this variational principle.

Using the more concise language of the variational bicomplex (notations and sign differ), if the Euler-Lagrange form $EL$ is defined by

$$ d_V\mathcal{L} = EL + d_H\Theta_{\mathcal{L}} $$

then it seems that $EL = d\Theta_{\mathcal{L}}$ which I can't explain.

[1] : Multisymplectic geometry, variational integrators, and nonlinear PDEs - Jerrold E. Marsden, George W. Patrick, Steve Shkoller

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