U.S. Particle Accelerator School

Integrable Particle Dynamics in Accelerators


Northern Illinois University and UT-Battelle

Course Name:

Integrable Particle Dynamics in Accelerators


Sergei Nagaitsev and Timofey Zolkin, Fermillab         Course Materials

Purpose and Audience
The purpose of this course is to introduce the students to integrable particle dynamics in circular accelerators. It is designed for graduate students pursuing accelerator physics as a career or having interest in learning this subject of accelerator physics.  The course is also appropriate for scientists and engineers working in accelerator-related fields who wish to broaden their background.

Classical mechanics, special relativity, electrodynamics, and physical or engineering mathematics, all at entrance graduate level, and the USPAS course “Accelerator Physics” (graduate level) or “Accelerator Fundamentals” with a strong mathematical background or equivalent.

It is the responsibility of the student to ensure that he or she meets the course prerequisites or has equivalent experience.

Upon completion of this course, the students are expected to understand the basic principles that underpin the physics of integrable particle dynamics in particle accelerators. They will have learned the notion of integrability as referring to the existence of the adequate number of invariants or constants of motion.  Applying this knowledge, they will then have developed an insight into the mechanisms of chaotic and non-chaotic particle motion in accelerators, action-phase variables, examples of nonlinear focusing systems leading to integrable motion, and nonlinear beam-beam effects.

Instructional Method
This course includes a series of lectures and exercise sessions.  Homework problems will be assigned daily which will be graded and answers provided in the exercise session the following day. There will be an in-class open-book final exam at the conclusion of the course.

Course Content
It has been known since the 19th century that non-integrable systems constitute the vast majority of all real-world dynamical systems, including particle accelerators. In accelerators, any arbitrary nonlinearity (sextupoles, octupoles, etc) is non-integrable. As dynamical systems, modern accelerators are characterized by an infinite number of resonances, chaotic motion around unstable points, diffusion, particle losses, and beam blow-up.  The distinction between integrable and non-integrable dynamical systems thus has the qualitative implication of regular motion vs. chaotic motion and hence is an intrinsic property, not just a matter of whether a system can be explicitly integrated in exact analytic form. 

The course will start with a Hamiltonian description of particle accelerators, examples of a linear integrable system and the Courant-Snyder invariant. Following this introduction, several examples of well-known integrable dynamical systems will be considered. The core of this course will be devoted to the following topics: nonlinear accelerator mappings, McMillan and Henon 1D mappings, accelerator-specific issues associated with nonlinear integrable systems and mappings, 2D and 3D systems realizable with static fields in vacuum, and including axially symmetric beam-beam systems and electron lenses, nonlinear particle traps and N-particle systems.

Reading Requirements
(to be provided by the USPAS) Jorge V. José and Eugene J. Saletan, “Classical Dynamics: A Contemporary Approach” (1998) Cambridge University Press.

Suggested Reading
SY Lee “Accelerator Physics”

A. Lichtenberg and M. Lieberman, “Regular and Chaotic Dynamics”

B. Chirikov “A Universal Instability of Many-Dimensional Oscillator Systems”, PHYSICS REPORTS (Review Section of Physics Letters) 52, No. 5 (1979) 263-379.

E. M. McMillan, “A problem in the stability of periodic systems”, Topics in Modern Physics. A Tribute to E. U. Condon, ed. W.E. Britton and H. Odabasi (Colorado Associated University Press, Boulder), pp 219-244 (1971).

Hénon, M. "Numerical Study of Quadratic Area-Preserving Mappings." Quart. Appl. Math. 27, 291-312, 1969.

Credit Requirements
Students will be evaluated based on the following performances: Final exam (50%), Homework assignments (30%) and class participation (20%).

Northern Illinois University course number: PHYSICS 790D - Special Topics in Physics - Beam Physics
Indiana University course number: Physics 671, Advanced Topics in Accelerator Physics
Michigan State University course number: PHY 963, "U.S. Particle Accelerator School"
MIT course number: 8.790, "Accelerator Physics"