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If we consider a classical action from r0 to r in terms of t(time), then for the semiclassical calculation, I can consider a small wavepacket and make an expansion around r0_bar.

Based on classical mechanics, we can have this formula S(r,r0;t)=S(r,r0_bar:t)-(r0-r0_bar)p_s, here p_s can be seen as the initial momentum of the orbit of S(r,r0_bar:t), this is

indeed out of any problem. But I want to check this formula with standard map or kicked top, but I find I can not make sense in this model. As the discrete characteristics, action difference is large as the two orbits should go to the same final position r in the same time. But this formula should fit for the Symplectic maps, so what is wrong with my idea?

I am working on this seemingly very easy job for quite a few days, but now it seems hopeless to find the reason. Maybe I have a lack of some knowledge here. So I ask for your help and very glad to hear some good suggestions from you. I give the related paper for the background with the formula(15) and (24).