Non-linear dynamics: phenomenology, tools and methods

This lecture covers the variables in Hamiltonian and Lagrangian formulations, the concept of Phase Space and Configuration Space, Action-Angle variables, Hamilton and Hamiltonian principles, Lagrange function, canonical variables, Hamiltonian for electromagnetic fields, Lie operators, Poisson brackets, Lie transformations, and applications in beam dynamics. The instructor explains the significance of Lie transforms in describing nonlinear elements, symplecticity, and the generation of transfer maps. The lecture delves into the analysis of Hamiltonians for machine elements, the derivation of invariants, and the application of Lie transforms in beam-beam interactions. The discussion extends to the calculation of tunes, chromaticities, and the importance of normal forms in nonlinear systems.