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Coffee_

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It's about equation (6.5) I'm not entirely getting the reasoning explained by the author so I came up with the following, can anyone confirm or refute. One way to look at equation (6.5) would be:

We create variations on the ##q## variables, in the form of ##\delta q(t)##. Since ##Q=Q(q,p,t)## the former variation induces a unique variation ##\delta Q(t)##. Since I want both of the action integrals to make sense, both ##\delta q(t) = \delta Q(t) ## have to be ##0## on the end points. This is not so subtle because technically a variation in ##q(t)## could change ##p(t)## in a non trivial way and ##\delta Q(t)## wouldn't be zero on the end points. However I have to make thsi a condition for both integrals over the Lagrangians to be identifiable as the action.

Under these conditions it is clear that the extra term with the ##F## vanishes.

Is this totally wrong, is this partially correct?