A Reduction Method for Higher Order Variational Equations of Hamiltonian Systems

Abstract. Let k be a differential field and let [A] : Y ′ = AY be a linear differential system where A ∈ Mat(n , k). We say that A is in a reduced form if A ∈ g(k̄) where g is the Lie algebra of [A] and k̄ denotes the algebraic closure of k. We owe the existence of such reduced forms to a result due to Kolchin and Kovacic [Kov71]. This paper is devoted to the study of reduced forms, of (higher order) variational equations along a particular solution of a complex analytical hamiltonian system X. Using a previous result [AW], we will assume that the first order variational equation has an abelian Lie algebra so that, at first order, there are no Galoisian obstructions to Liouville integrability. We give a strategy to (partially) reduce the variational equations at order m+1 if the variational equations at order m are already in a reduced form and their Lie algebra is abelian. Our procedure stops when we meet obstructions to the meromorphic integrability of X. We make strong use both of the lower block triangular structure of the variational equations and of the notion of associated Lie algebra of a linear differential system (based on the works of Wei and Norman in [WN63]). Obstructions to integrability appear when at some step we obtain a non-trivial commutator between a diagonal element and a nilpotent (subdiagonal) element of the associated Lie algebra. We use our method coupled with a reasoning on polylogarithms to give a new and systematic proof of the non-integrability of the Hénon-Heiles system. We conjecture that our method is not only a partial reduction procedure but a complete reduction algorithm. In the context of complex Hamiltonian systems, this would mean that our method would be an effective version of the MoralesRamis-Simó theorem.