Is there any relation between Galois solvability and integrability of hamiltonian systems?

Question

Galois theory provides a method and formalism to study solutions of polynomial equations and solvability.

Dynamical Hamiltonian systems have a somewhat similar concept of integrability.

Since many connections or reductions exist between differential equations and polynomials (eg by a Fourier tranform or others..)

Is there any connection or relation between these concepts of solvability and integrability (at least for some cases)?

PS. A related question

@Raskolnikov, checked it out, however did not see relations between solvability and integrability (although it has some info on "Liouvile extension") am i missing sth? — Nikos M., Jun 16 '14 at 12:08
@Raskolnikov, i guess Picard-Vessiot theory, provides a relation between (differential) Galois solvability and Liouvillian extension of the underlying differential field, is this all? — Nikos M., Jun 16 '14 at 12:21
There is something called Morales–Ramis theory, which I think might be what you're looking for. (But I don't know much about it, unfortunately, so I'll just leave this as a comment.) — Hans Lundmark, Jun 16 '14 at 12:53
@HansLundmark, found the original Moralez-Ramis paper (circa 2001), from a quick look it is an answer to the question — Nikos M., Jun 16 '14 at 13:14
@HansLundmark, why dont you post an answer based on the references (maybe a little summary) and it will be accepted. i would expect you to post an answer, since noone seems to post one but your references provide an answer — Nikos M., Mar 10 '15 at 16:45

score 3 · Accepted Answer · answered Mar 13 '15 at 17:41

The so-called Morales–Ramis theory seems to be what you are looking for. Several articles, plus a draft version of the book Differential Galois Theory and Non-integrability of Hamiltonian Systems can be downloaded from the homepage of Prof. Juan J. Morales-Ruiz.

I can't say that I know very much about this subject, but the little that I do know, I have learned from Maria Przybylska, who has written many papers together with Andrzej Maciejewski where they apply this theory. For example, they have found some polynomial potentials $V(x,y)$ such that the Hamiltonian system generated by $H=\frac12(p_x^2+p_y^2)+V(x,y)$ has an additional integral of motion (independent of $H$) which is of degree four in the momenta $p_x$ and $p_y$. Even though the polynomial $V(x,y)$ can be simple-looking, the second integral of motion may be completely monstrous (many pages of Mathematica output), and it's nothing that you would ever stumble upon just by blind trial and error, but using Morales–Ramis theory it's possible to systematically rule out non-integrable cases and then find these rare and very special integrable potentials.

Is there any relation between Galois solvability and integrability of hamiltonian systems?

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