Galoisian obstructions to integrability of Hamiltonian systems

Abstract

An inconvenience of all the known galoisian formulations of Ziglin's non-integrability theory is the Fuchsian condition at the singular points of the variational equations. We avoid this restriction. Moreover we prove that a necessary condition for meromorphic complete integrability (in Liouville sense) is that the identity component of the Galois group of the variational equation (in the complex domain) must be abelian. We test the efficacy of these new approaches on some examples. We will give some non academic applications in two following papers.

Computer assisted studies in dynamical systems

“…Полагая, что t ∈ C -множество комплексных чисел, получаем, что реше-ние z определяет риманову поверхность Γ. Мы можем рассмотреть вариации первого порядка гамильтонова векторного поля вдоль Γ. С этими (линейными) уравнениями связана группа Галуа G расширения Пикара-Вессио основного поля функций (определенного коэффициентами ва-риаций в Γ). В [13] получен следующий ключевой результат, известный как теорема Моралеса-Рамиса.…”

La non-intégrabilité méromorphe du problème plan des trois corps

“…Dans le cas de masseségales, le même résultat aété obtenu par D. Boucherà l'aide du théorème de non-intégrabilité de Moralis-Ramis [3].…”

Differential Galois Groups and Representation of Quivers for Seismic Models With Constant Hessian of Square of Slowness

“…Morales-Ramis theory relates integrability of Hamiltonian systems in the Liouville sense with integrability Picard-Vessiot theory in terms of differential Galois theory. The following theorem is known as Morales-Ramis theorem, see [32][33][34][35].…”

“…Following [32][33][34][35], we say that H in integrable by terms of rational functions if we can find a complete set of integrals within the family of rational functions. Respectively, we can say that H is integrable by terms of meromorphic functions if we can find a complete set of integrals within the family of meromorphic functions.…”