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).
I'ev never used such approach. I attached two papers, which could usful. In particular, first paragraph on p.31 in PE.pdf and last paragraph on p.345 in EPJD.pdf
I think we should have a very clear understanding the variation method here, because the S(r,r0;t)=S(r,r0_bar:t)-(r0-r0_bar)p_s is a basic relation in classical mechanics, but the derivation come from the variation which means there only have one real orbit, but in the paper I want to study is that S(r,r0;t) can be taken as a function for different r0, but actually there have many different orbits with the classical requirement from the r0 to r in a given same time in terms of different related momentum, I also check that. So it is a problem for me that how to understand this expression S(r,r0;t)=S(r,r0_bar:t)-(r0-r0_bar)p_s? S(r,r0;t) actually is the Hamilton principle function in the standard textbook. I think indeed my understanding have something wrong. much welcome anyone having good idea in this problem.
Dear Manuel Morales, maybe you can check this formula with your idea, if you have some interest, ok? such as using the very simple model-kicked rotator. I am glad to hear your good news and I am very pleased to see that result. Best!
I basically figure out the results and I think any complication question can be clarified if we could find the most simple case to understand. I welcome anyone who have the deep understanding about Hamiltonian dynamics to discuss with me.Thank you!
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