(with M Ayoul) Galoisian obstructions to non-Hmailtonian integrability, Comptes Rendus Mathématiques, Volume 348, Issues 23–24, December 2010, Pages 1323-1326.
This paper, despite being a short paper in CRAS, contains a rather strong result: the differential Galois group of the variational equation of any order along a solution of an analytic integrable dynamical system is virtually Abelian. Our result extends the theorem of Morales-Ramis-Simo to the non-Hamiltonian case, and its proof is also based on Morales-Ramis-Simo’ theorem. It also shows that the Hamiltonian condition (and arguments based on it) of Morales-Ramis-Simo theorem are in fact superfluous, and one could write a proof from scratch without any “Hamiltonian” word in it. Since Morales-Ramis-Simo already laid out the foundations, instead of writing such a proof, we simply used their result, and a Hamiltonian – nonHamiltonian correspondence to prove our theorem. Michael Ayoul was a student of mine, and this probelm was his PhD project.
- # A. Sergyeyev, Coupling constant metamorphosis as an integrability-preserving transformation for general finite-dimensional dynamical systems and ODEs, preprint arXiv:1008.1575, 2010.
- W. Li, S. Shi, Galoisian obstruction to the integrability of general dynamical systems, J. Diff. Equations, 2012.
- Andrzej J. Maciejewski, Maria Przybylska, Differential Galois theory and Integrability, International Journal of Geometric Methods in Modern Physics, Volume: 6, Issue: 8 (2009) pp. 1357-1390.
- G Casale, Morales-Ramis Theorems via Malgrange pseudogroup, Annales de l’institut Fourier, 59 no. 7 (2009), p. 2593-2610.
- ^# NT Zung, Nondegenerate singularities of integrable non-Hamiltonian systems, preprint arXiv:1108.3551, 2011.
An old article which contains some ideas about topological obstructions to integrability:
(with Tit Bau) Singularities of integrable and near-integrable Hamiltonian systems. J. Nonlinear Sci. 7 (1997), no. 1, 1–7