Differential Galois Theory And Non-integrability Of Hamiltonian Systems (draft)

Winner of the 1998 Ferran Sunyer i Balaguer Prize 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. In addition to the theory, the book also presents several important applications, including homogeneous potentials, Bianchi IX cosmological model, three-body problem, Hnon-Heiles system, etc. The book is based on the original joint research of the author with J.M. Peris, J.P. Ramis and C. Sim, 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. Series: Progress in Mathematics, Vol. 179