As the title says, I'm looking for an explanation on how to apply canonical transformations when using the covariant phase space formalism. I'm familiar with the topic, but I haven't found a good explanation for canonical transformations. The only one I could find is in the book by DeWitt, but it is explained on page 700 on the second volume and I can't even get past the first volume, it's a bit heavy on the notation and the generalizations to include Grassmann variables and all that.
My Attempt:
In the covariant phase space formalism we start with an action principle
\begin{equation} S=\int _M L\tag{1} \end{equation} that integrates a $D$-form in spacetime over a manifold $M$. A variation of the action respect to the dynamical degrees of freedom, collectively denoted by $\phi^a$ is
\begin{equation} \delta S =\int _M E_a [\phi]\delta \phi^a+\int_{\partial M}\theta[\phi,\delta\phi]\tag{2} \end{equation}
where we used $\delta L=E_a\delta\phi^a+d\theta$. $\theta$ is the pre-symplectic potential and $E_a$ are the Euler-Lagrange equations. The equations of motion are obtained by demanding $\delta S=0$ to get
\begin{equation} E_a =0.\tag{3} \end{equation}
The presymplectic potential needs to satisfy boundary conditions for the theory to be well defined, so we get \begin{equation} \theta |_{\Gamma}=dC+F\tag{4} \end{equation}
where $C$ is an allowed corner term and $F$ is a flux term that exists for null boundaries and represents the failure of the symplectic form to be zero at the boundaries.
With the presymplectic potential we can build a presymplectic current as
\begin{equation} \omega[\delta\phi_1,\delta\phi_2]=\delta\phi_1 \theta[\phi,\delta\phi_2]-\delta\phi_2\theta[\phi,\delta\phi_1].\tag{5} \end{equation}
Integrating this over a Cauchy surface $\Sigma$ we get the presymplectic form $\tilde{\Omega}$
\begin{equation} \tilde{\Omega}=\int_\Sigma \omega.\tag{6} \end{equation}
My idea would be to first obtain a generalized version of Hamilton's equations. That would look like
\begin{equation} \delta H_s =\tilde{\Omega}\big[\delta \phi,\delta_s \phi\big]\tag{7} \end{equation}
which means that the conserved charge $H_s$ for the symmetry transformation $s$ generates the change $\delta_s \phi$. If we take $s$ to be time translations, this is exactly Hamilton's equations. Then, the idea would be to find transformations on the fields $\phi^a$ that keep this equation unchanged.
Alternatively, $H_s$ might also be thought of as the infinitesimal generator of the canonical transformation $s$, although I'm not sure if that generally makes sense. I'm stuck.
