U.S. Particle Accelerator School

Classical Mechanics and EM for Accelerators and Beams course

Sponsoring University:

College of William and Mary


Classical Mechanics and EM for Accelerators and Beams


Helmut Wiedemann, Stanford University

This course is designed to guide the student through some advanced treatments of particle dynamics. We start with the Vlasov equation to describe the evolution of an ensemble of particles, while elucidating Liouville’s theorem, symplecticity and damping. Simultaneously, we arrive at the need for Hamilton’s equations which we derive from general principles. Conjugate coordinates and canonical transformations are introduced as a means to extract particle dynamics free of already known quantities. A canonical transformations in particle beam dynamics, for example, can reveal the structure of one-dimensional and coupling resonances. Proceeding along this line, we enter Hamiltonian nonlinear beam dynamics and derive from Hamiltonian Perturbation theory higher-order tune shift, which is of particular interest in accelerators employing sextupole fields. Expanding on the Vlasov equation we introduce the Fokker-Plank equation to deal with statistical processes.

Interaction of particles and electromagnetic fields is introduced on a very basic level to extract the character of this interaction. Emission/absorption of EM energy by charged particles can be described either as a Cherenkov or Compton interaction. Emission of synchrotron radiation is introduced as a consequence of the finite velocity of light. To transform from the particle system to the laboratory system we introduce the Lorentz transformation and 4-vector analysis. Relativistic Doppler effect and forward collimation of synchrotron radiation from high-energy charged particles results automatically from such transformations. We derive the nature of undulator, wiggler and bending magnet radiation and discuss briefly the dynamics of a free electron laser and of a single pass FEL. Finally, we discuss some more unconventional femtosecond x-ray sources by Thomson scattering from femtosecond electron bunches, or through interaction of such bunches with periodic structures on a nano- or crystalline scale and discuss the generation of coherent transition radiation. Prerequisites: Basic understanding of beam physics, general university course on classical mechanics and EM.  Textbook to be provided: Particle Accelerator Physics I & II by Helmut Wiedemann (2nd edition, 2nd printing, Springer-Verlag, 2003).