This book is devoted to the relation between two different concepts of
integrability: the complete integrability of complex analytical
Hamiltonian systems and the integrability of complex analytical linear
differential equations. For linear differential equations, integrability
is made precise within the framework of differential Galois theory. The
connection of these two integrability notions is given by the
variational equation (i.e. linearized equation) along a particular
integral curve of the Hamiltonian system. The underlying heuristic idea,
which motivated the main results presented in this monograph, is that a
necessary condition for the integrability of a Hamiltonian system is the
integrability of the variational equation along any of its particular
integral curves. This idea led to the algebraic non-integrability
criteria for Hamiltonian systems. These criteria can be considered as
generalizations of classical non-integrability results by Poincaré and
Liapunov, as well as more recent results by Ziglin and Yoshida. Thus, by
means of the differential Galois theory it is not only possible to
understand all these approaches in a unified way but also to improve
them. Several important applications are also included: homogeneous
potentials, Bianchi IX cosmological model, three-body problem,
Hénon-Heiles system, etc.
The book is based on the original joint research of the author with J.M.
Peris, J.P. Ramis and C. Simó, but an effort was made to present these
achievements in their logical order rather than their historical one.
The necessary background on differential Galois theory and Hamiltonian
systems is included, and several new problems and conjectures which open
new lines of research are proposed.
