= Solution
The defining relations for a <type-two generating function for a canonical transformation>[type-2 generating function] give
$$
d(F_2-P\cdot Q)=p\cdot dq-P\cdot dQ
+\frac{\partial F_2}{\partial t}dt.
$$
Hence
$$
p\cdot dq-Hdt
=P\cdot dQ-\left(H+\frac{\partial F_2}{\partial t}\right)dt
+d(F_2-P\cdot Q).
$$
The exact differential changes the action only by an endpoint term. Thus the transformed variational principle has canonical one-form $P\cdot dQ-Kdt$ and yields Hamilton's equations in $(Q,P)$ with
$$
\boxed{K=H+\partial_tF_2}.
$$
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