A Reduction Method for Higher Order Variational
Equations of Hamiltonian Systems
A. Aparicio Monforte and J.-A. Weil
Abstract. Let k be a diﬀerential ﬁeld and let [A] : Y = A Y be a linear
diﬀerential system where A ∈ Mat(n , k). We say that A is in a reduced form
if A ∈
k where g is the Lie algebra of [A] and
k 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 H´ enon-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 Morales-
Ramis-Sim´ o theorem.
Let (k , ) be a diﬀerential ﬁeld and let [A] : Y = AY be a linear diﬀerential
system with A ∈ Mn(k). We say that the system is in reduced form if its matrix
can be decomposed as A =
αiAi where αi ∈ k and Ai ∈ Lie(Y = AY ), the
Lie algebra of the diﬀerential Galois group of [A].
2010 Mathematics Subject Classiﬁcation. Primary 37J30, 34A05, 68W30, 34M15, 34M25,
34Mxx, 20Gxx ; Secondary 20G45, 32G81, 34M05, 37K10, 17B80 .
Key words and phrases. Diﬀerential Galois Theory, Integrability, Dynamical Systems.
The ﬁrst author was supported by a Grant from the Region Limousin (France).
c 0000 (copyright holder)
Volume 549, 2011