TITLE: Non-integrability of Hamiltonian systems through high order variational 
equations: Summary of results and examples. 

AUTHORS: 
R. Martinez
Dept. de Matematiques, Univ. Autonoma de Barcelona
Barcelona, Catalunya, Spain 
E-mail:reginamb@mat.uab.cat

C. Simo
Dept. de Matematica Aplicada i Analisi, Univ. de Barcelona
Barcelona, Catalunya, Spain
E-mail:carles@maia.ub.es

ABSTRACT:
This paper deals with non-integrability criteria, based on differential Galois 
theory and requiring the use of higher order variational equations. A general
methodology is presented to deal with these problems. We display a family
of Hamiltonian systems which require the use of order k variational equations,
for arbitrary values of k, to prove non-integrability. Moreover, using third
order variational equations we prove the non-integrability of a non-linear
spring-pendulum problem for the values of the parameter that can not be decided
using first order variational equations.


Preprint submitted to Regular and Chaotic Dynamics, 28 November 2008.

Published in Regular and Chaotic Dynamics, 14 (2009) 323-348.