Based on lectures given at an advanced course on integrable systems at the Centre de Recerca Matemàtica in Barcelona, these lecture notes address three major aspects of integrable systems: obstructions to integrability from differential Galois theory; the description of singularities of integrable systems on the basis of their relation to bi-Hamiltonian systems; and the generalization of integrable systems to the non-Hamiltonian settings. All three sections were written by top experts in their respective fields.

Native to actual problem-solving challenges in mechanics, the topic of integrable systems is currently at the crossroads of several disciplines in pure and applied mathematics, and also has important interactions with physics. The study of integrable systems also actively employs methods from differential geometry. Moreover, it is extremely important in symplectic geometry and Hamiltonian dynamics, and has strong correlations with mathematical physics, Lie theory and algebraic geometry (including mirror symmetry). As such, the book will appeal to experts with a wide range of backgrounds.

Provides a clear introduction to Differential Galois Theory and to Picard-Vessiot Theory Establishes, as a first book, a connection between Singularities of bi-Hamiltonian systems, stability analysis, and Poisson pencils Shows how to apply the tools used in integrable Hamiltonian systems to integrable non-Hamiltonian systems, with applications in Control Theory, economics, and biology

Autorentext

Juan J. Morales-Ruiz is Professor of Mathematics at Universidad Politécnica de Madrid.

Alexey Bolsinov is Reader in Mathematics at Loughborough University in Leicestershire.

Nguyen Tien Zung is Professor of Mathematics at University of Toulouse.

Inhalt

Integrable Systems and Differential Galois Theory.- Singularities of bi-Hamiltonian Systems and Stability Analysis.- Geometry of Integrable non-Hamiltonian Systems.