Differential Galois Theory and Non-Integrability of Hamiltonian Systems

Juan J. Morales Ruiz is Professor at the Universidad Politécnica de Madrid, Spain.

• 1 Introduction.- 2 Differential Galois Theory.- 2.1 Algebraic groups.- 2.2 Classical approach.- 2.3 Meromorphic connections.- 2.4 The Tannakian approach.- 2.5 Stokes multipliers.- 2.6 Coverings and differential Galois groups.- 2.7 Kovacic’s algorithm.- 2.8 Examples.- 3 Hamiltonian Systems.- 3.1 Definitions.- 3.2 Complete integrability.- 3.3 Three non-integrability theorems.- 3.4 Some properties of Poisson algebras.- 4 Non-integrability Theorems.- 4.1 Variational equations.- 4.2 Main results.- 4.3 Examples.- 5 Three Models.- 5.1 Homogeneous potentials.- 5.2 The Bianchi IX cosmological model.- 5.3 Sitnikov’s Three-Body Problem.- 6 An Application of the Lamé Equation.- 6.1 Computation of the potentials.- 6.2 Non-integrability criterion.- 6.3 Examples.- 6.4 The homogeneous Hénon-Heiles potential.- 7 A Connection with Chaotic Dynamics.- 7.1 Grotta-Ragazzo interpretation of Lerman’s theorem.- 7.2 Differential Galois approach.- 7.3 Example.- 8 Complementary Results and Conjectures.- 8.1 Two additional applications.- 8.2 A conjecture about the dynamic.- 8.3 Higher-order variational equations.- A Meromorphic Bundles.- B Galois Groups and Finite Coverings.- C Connections with Structure Group.

"...[an] account of recent work of the author and co-workers on obstructions to the complete integrability of complex Hamiltonian systems. The methods are of considerable importance to practitioners... The book provides all the needed background...and presents concrete examples in considerable detail... The final chapter...includes a fascinating account of work-in-progress by the author and his collaborators... Of particular interest...is the program of extending the differential Galois theory to higher-order variational equations... [an] excellent introduction to non-integrability methods in Hamiltonian mechanics [that] brings the reader to the forefront of research in the area. The inclusion of a large number of worked-out examples, many of wide applied interest, is commendable. There are many historical references, and an extensive bibliography." --Mathematical Reviews