U.S. Particle Accelerator School

U.S. Particle Accelerator School

Education in Beam Physics and Accelerator Technology

University of California, Berkeley

Classical Mechanics and EM in Accelerator Physics

Gennady Stupakov, SLAC and Marco Venturini, LBNL

THIS COURSE HAS BEEN CANCELLED

**Purpose and Audience **

Classical mechanics and electromagnetism play an important role in the dynamics of dense particle beams. This course is designed to introduce students to advanced concepts of beam dynamics and emission of radiation from relativistic electrons.

**Prerequisites
**A general university course on Classical Mechanics and EM.

**Objectives **

The main objective of this course is to present the most relevant accelerator physics concepts of classical mechanics and EM. The selected topics will be covered in depth and illustrated with computer simulations. An emphasis will be made not so much on formal derivations of equations but on understanding of the underlying physical phenomena.

**Instructional Method **

The format will be primarily lectures supplemented with overheads and PowerPoint presentations. Morning lectures will total 3 hrs. per class day (30 hrs.). In addition, afternoon lectures and/or computer lab time will total a minimum of 2 hrs. per class day (20 hrs.). The instructor will be available at all times. Questions and discussions are encouraged.

**Course Content **

This course focuses on several topics of classical mechanics and electrodynamics of particular importance for accelerator physics. We will start with Lagrangian formulation of equations of motion and Hamilton's equations. Canonical transformations and action-angle variables will be introduced upon which we will develop the Hamiltonian theory of perturbations and apply it to nonlinear resonances arising in circular accelerators. The mechanism of resonance overlapping and transition to stochastic motion will be discussed.

The concept of adiabatic invariants in classical mechanics will be illustrated with examples from beam dynamics. We will briefly review Lorentz transformations and the Lienard-Wiechert potentials as a basis for calculation of radiation processes in electrodynamics. We will consider self-fields of relativistic beams and study simple examples of wakefields resulting from interaction of the beam with environment. A detailed study of properties of synchrotron radiation will be presented. Other types of radiation such as undulator radiation, transition radiation, and Compton scattering will also be considered with emphasis on properties of spatial and temporal coherence and formation length of the radiation. We will also take a look at different mechanisms of laser interaction with relativistic beams and discuss the fundamental Lawson-Woodward theorem. The intimate relation between radiation and acceleration will be described and illustrated with examples of 'inverse radiation' acceleration schemes.

**Reading Requirements **

References: B. W. Montague, Basic Hamiltonian Mechanics, printed in the Proceedings of the CERN Accelerator School, CERN 95-06; E. J. N. Wilson, Nonlinear Resonances, printed in the Proceedings of the CERN Accelerator School, CERN 95-06.

*(To be provided by the USPAS)*: "Classical Electrodynamics" by J. D. Jackson, (1999) Wiley publishers.

**Credit Requirements **

Students will be evaluated based on performance: final exam (40% of final grade) and assignments (60% of final grade).