Is there a polynomial Hamiltonian vector field with a finite number of periodic orbits?
Please see this MO question:
https://mathoverflow.net/questions/210996/an-algebraics-hamiltonian-vector-field-with-a-finite-number-of-periodic-orbits2
Thank you
Related to your query, I suggest you follow : https://mathoverflow.net/questions/155590/an-algebraic-hamiltonian-vector-field-with-a-finite-number-of-periodic-orbits1?noredirect=1&lq=1
https://mathoverflow.net/questions/157966/the-perturbation-of-non-hamiltonian-algebraic-vector-fields
Muhammad Ali Thank you very much for your attention to my question. The links you shared are the link of my questions in MO. The first one is related to my question. it is about a particular Hamiltonian in R^4. the answer to that question considers a more general case:i.e special hamiltonians on the tangent bundle of manifolds. But my main question is stil unsolved:"is there an algebraic hamiltonian vector field on R^4 with the following property: The corresponding Hamiltonian vector field X_H possess a finite (but non zero) number of closed orbits"
Namely: if we remove the set S of singularities of vectorfield from R^4, we would have a foliation of R^setminus S. In our question we require that the number of closed leavs of the foliation would be finite but non zero.
I presented this question in MO since many years ago but it is stil unsolved. I wonder which kind of hamiltonian or symplectic technics can be applied for this question.
The second link you mention is not directly related to my question. But it is possible one find an indirect but very helpful relation. I do not know.
@Ali Taghavi
Please be specific.
Write a model Hamiltonian , so that
I can try for you ,since it was question as per your statement.
Biswanath Rath The Hamiltonian vector field associated to H(x,y,z,w) is simply the following vector field:
x'=\partial H/\partial z
y'=\partial H/\partial w
z'=-\partial H/\partial x
w'=-\partial H/\partial y
So our question is the following: Is there a polynomial H such that the above vector field possess (exactly) k periodic orbit for a finite number k different zero?
The motivation for this question: In dimension 2 (rather than dim. 4) this situation can not occured(we have a bound of periodic orbits, if there is any)
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