I am working on the Hamiltonian formulation of the Einstein equations of motion in General Relativity, where the aim is to find the Hamiltonian generating the dynamics from the Einstein equations (where one already has the Lagrangian).
On the phase space of a manifold $M$, that is the cotangent bundle $T^\ast M,$ one can always define a canonical symplectic form $\omega.$ When the manifold is normal coordinates, as it is the case for example for a mechanical system of $N$ bodies, we then have for the Hamiltonian of the system the relation $$\omega(T, \cdot)=\mathrm{d}H,$$ where $T$ is the vector field whose flow provides solutions to the equations of motion of the system.
My question is: why exactly is it important to have the canonical symplectic form in this setting when looking for the appropriate Hamiltonian, where our manifold is the space of all (gravitational) fields $\varphi_{\mu\nu}$ instead of some coordinate points? Is there an analogy to the relation given above, so I can find the Hamiltonian given the symplectic form and the dynamics from the Einstein equations?
