Contemporary Mathematics

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 ∈

g(¯)

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.

1. Introduction

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 =

∑d

i=1

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)

1

Contemporary Mathematics

Volume 549, 2011

1

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 ∈

g(¯)

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.

1. Introduction

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 =

∑d

i=1

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)

1

Contemporary Mathematics

Volume 549, 2011

1