U.S. Particle Accelerator School
U.S. Particle Accelerator School
Education in Beam Physics and Accelerator Technology

Integrable Particle Dynamics in Accelerators


Northern Illinois University

Course Name:

Integrable Particle Dynamics in Accelerators
This class is limited to 15 students


Timofey Zolkin and Jeff Eldred, Fermilab

Purpose and Audience
The purpose of this course is to introduce the students to the concept of integrable dynamics and its application to circular charged particle accelerators. Accelerator systems designed on the basis of integrable nonlinear focusing systems promise enhanced stability properties leading to desirable features relative to machines based on linear optics. This course will overview advances in this field stimulating the IOTA project at Fermilab.  The course is designed for graduate students pursuing accelerator physics as a career or having interest in learning this subject. It is also appropriate for scientists and engineers working in accelerator-related fields who wish to broaden their background.

Classical mechanics, special relativity, electrodynamics, and applied mathematical methods for scientists and engineers, all at entrance graduate level, are required. Familiarity with accelerator science at the level of the USPAS course “Accelerator Physics” (graduate level) or “Accelerator Fundamentals” (undergraduate level), or equivalent experience, is required.  It is recommended that prospective students have a strong background in applied mathematical methods used in accelerator physics and nonlinear dynamics.

It is the responsibility of the student to ensure that they meet the course prerequisites or have 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 learn the notion of integrability as referring to the existence of the adequate number of invariants or constants of motion.  Applying this knowledge, they will develop an insight into the mechanisms of chaotic and non-chaotic particle motion in accelerators, action-angle variables, examples of nonlinear focusing systems leading to integrable motion, and nonlinear beam-beam effects. Both analytical and numerical methods will be covered.

Instructional Method
This course includes a series of lectures and exercise sessions.  Homework problems will be assigned daily which will be graded and solutions will be reviewed in exercise session the following day. There will be an in-class, open-note 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. Additionally, since integrable systems are usually fine tuned in a space of parameters, the concept of near-integrability and structural stability will be discussed.

The course will start with a brief review of differential and difference equations in a context of accelerator physics. Starting with a Hamiltonian description of harmonic oscillator and Courant-Snyder formalism for linear mappings, we will proceed to nonlinear generalizations associated with sextupole and octupole lenses in particle accelerators. Following this introduction, several examples of well-known integrable, near-integrable and chaotic 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.

Methodologies developed will illustrate the nonlinear integrable optic system forming the basis of the ongoing IOTA experiment at Fermilab. The course will connect examples to this project and help frame and motivate benefits expected relative to linear optics.

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

Suggested Reading
- Accelerator Physics, fourth edition by S.Y. Lee (World Scientific 2019)
- Regular and Chaotic Dynamics, second edition by A.J. Lichtenberg and M.A. Lieberman (Springer 1992)
- A Universal Instability of Many-Dimensional Oscillator Systems by Boris Chirikov (PHYSICS REPORTS, Review Section of Physics Letters, 52, No. 5, 263-379, 1979).
- A problem in the stability of periodic systems by E.M. McMillan, Topics in Modern Physics, A Tribute to E. U. Condon, ed. W.E. Brittin and H. Odabasi (Colorado Associated University Press, Boulder, pp 219-244, 1971).
- Numerical Study of Quadratic Area-Preserving Mappings by M. Henon (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%).

USPAS Computer Requirements
There will be no Computer Lab and all participants are required to bring their own portable computer to access online course notes and computer resources. This can be a laptop or a tablet with a sufficiently large screen and keyboard. Windows, Mac, and Linux-based systems that are wifi capable and have a standard web browser and mouse are all acceptable. You should have privileges for software installs. If you are unable to bring a computer, please contact uspas@fnal.gov ASAP to request a laptop loan. Very limited IT support and spare loaner laptops will be available during the session.

Northern Illinois University course number: PHYS 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