Integrable Dynamical Systems in Particle Accelerators Sergei Nagaitsev University of Chicago Particle motion in accelerators represents a fascinating dynamical system. Particle interactions with each other and with stationary and time-dependent fields make motion in accelerators quite complex and, in general, non-integrable. As a consequence, nearly all accelerators are limited in their performance by an on-set of chaotic particle behavior, instabilities and associated uncontrolled beam losses. Improved integrability in both electron and ion accelerators has been a long-sought-after goal of accelerator physicists. Recently, we have proposed several examples of integrable nonlinear accelerator focusing systems. This talk will review the physics of such dynamical systems and the range of their applications in accelerators. A proposed demonstration experiments at Fermilab will be also discussed.