Non-linear dynamics

This lecture covers contemporary methods, techniques, and tools in nonlinear dynamics, emphasizing practical applications over theoretical exercises. Starting from linear dynamics, topics progress to Lagrange and Hamiltonian formalism, Lie operators, nonlinear maps, symplecticity, and beam-beam effects. The instructor motivates the study of nonlinear dynamics in accelerators, highlighting the importance of canonical variables and the transition to nonlinear elements. The lecture explores the challenges of integrating nonlinear fields, stability analysis, and the use of simulations for realistic models. Symplectic integration methods, thin lens approximations, and the symplecticity of maps are discussed in detail, providing insights into accurate and efficient computations in accelerator physics.