# Poincaré-Cartan form and Legendre transformation from the variational principle

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 $$$$\tag{1}\label{contact} \forall \phi\in\Gamma(\pi), j^1(\phi)^*\Theta_{\mathcal L} = j^1(\phi)^*\mathcal L = {\mathcal L}\circ j^1(\phi)$$$$

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.