Obstructions to integrability (2010)

 

(with M Ayoul) Galoisian obstructions to non-Hmailtonian integrability, Comptes Rendus Mathématiques, Volume 348, Issues 23–24, December 2010, Pages 1323-1326.

Arxiv: http://arxiv.org/abs/0901.4586

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.

Cited in:

  1. # A. Sergyeyev, Coupling constant metamorphosis as an integrability-preserving transformation for general finite-dimensional dynamical systems and ODEs, preprint arXiv:1008.1575, 2010.
  2. W. Li, S. Shi, Galoisian obstruction to the integrability of general dynamical systems, J. Diff. Equations, 2012.
  3. 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.
  4. G Casale, Morales-Ramis Theorems via Malgrange pseudogroup, Annales de l’institut Fourier, 59 no. 7 (2009), p. 2593-2610.
  5. ^# 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


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