I am struggling to see how the symplectic action 1-form $\alpha_H$ on the loop space of a symplectic manifold $(M,\omega)$ is closed. It is defined by

$\alpha_H(Y) = \int_0^1 \omega(\dot{x}(t)-X_H,Y)dt$

with $Y \in T_{x(t)}M$ and $X_H$ a Hamiltonian vector field on $M$. Going the other way and obtaining it as the differential of the symplectic action is fine (and so I understand why it is closed from this point of view), but I am not sure how to calculate $d\alpha_H$ explicitly and have not seen it done anywhere. To be honest, calculations on loop space are confusing me a bit and it would be useful to know of a source where this is all worked out in detail.