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Jun 10 at 19:51 comment added user17945 Glad it was helpful. I was surprised to hear ChatGPT could answer this, but I've just tried, and yes it does indeed.
Jun 10 at 14:11 history bounty ended Dmitri Scheglov
Jun 10 at 7:24 vote accept Dmitri Scheglov
Jun 10 at 7:18 comment added Dmitri Scheglov Nice answer. When I asked chatgpt, it suggested $F(q_1,q_3)=F_1(q_1,q_2)+F_2(q_2_q_3)$, where $q_2=q_2(q_1, q_3)$ is a minimizer of the sum above. I checked it for a pair of randomly chosen linear symplectic transformations and it was the case indeed. Of course, as you mentioned, it will not work universally, as, say identity transformation does not have qq generating function at all.
Jun 9 at 20:10 history edited user17945 CC BY-SA 4.0
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Jun 9 at 5:56 history edited user17945 CC BY-SA 4.0
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Jun 9 at 4:26 history answered user17945 CC BY-SA 4.0