U.S. Particle Accelerator School

The Use of Hamiltonian & Lie Algebra Methods to Analyze & Design Accelerator Beamlines course

Sponsoring University:

University of Chicago


The Use of Hamiltonian & Lie Algebra Methods to Analyze & Design Accelerator Beamlines


John Irwin, SLAC

This course will be self-contained in that no knowledge of Hamiltonian or Lie methods will be assumed, though it would be helpful for the student to have experience with the Courant-Snyder approach to linear optics. Attention will be limited to single-particle dynamics. The course will start with intuitive introductions to Hamiltonian generators and their corresponding Lie operators. The three core relationships will be derived and/or described in detail: the product law of Lie operators to build beamlines, and the similarity and BCH laws to compose operators. The remainder of the course will be the application of these tools to beamline analysis and design, and monitoring the design process. Many examples will be presented and parallels with "generalized matrix methods" will often be developed. There will be an attempt to cover the following topics: 1) the complete electromagnetic hamiltonian in rectilinear & circular coordinate systems, 2) treatment of real elements (including solenoids) with full multipole complexity as well as edges and fringes, 3) integration techniques, including symplectic integrators and splitting and squaring, 4) misalignments (coordinate patches), 5) composition of elements, 6) beamline modules (achromats, final focus systems, beam shaping), 7) analysis of one-turn and beamline maps, 8) Dragt-Finn factorization, 9) resonance (action-angle) bases, 10) nPB tracking, 11) normal forms, 12) kick (Cremona) factorizations, and 13) classical synchrotron radiation.  Prerequisites: Accelerator physics, classical mechanics and electromagnetism.