U.S. Particle Accelerator School

U.S. Particle Accelerator School

Education in Beam Physics and Accelerator Technology

University of California, Davis

Integrable Particle Dynamics in Accelerators

Sergei Nagaitsev, Fermilab and the University of Chicago and Timofey Zolkin, Fermilab

**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.

**Prerequisites**

Classical mechanics, 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 math background or equivalent.

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

**Objectives**

Upon completion of this course, the students are expected to understand the basic principles that underline 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 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 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 form.

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

**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” (third edition) by S.Y. Lee**, **World Scientific (2012)

- “Regular and Chaotic Dynamics” by Allan Lichtenberg and Michael Lieberman, Springer (1992)

- “A Universal Instability of Many-Dimensional Oscillator Systems”, by B. Chirikov, 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).

**Credit Requirements**

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

**
UC Davis course number:** 163EDN671 Advanced Topics