Single particle dynamics and nonlinear resonances in circular accelerators
The purpose of this paper is to introduce the reader to single particle dynamics in circular accelerators with an emphasis on nonlinear resonances. We begin with the Hamiltonian and the equations of motion in the neighborhood of the design orbit. In the linear theory this yields linear betatron oscillations about a closed orbit. It is useful then to introduce the action-angle variables of the linear problem. Next we discuss the nonlinear terms which are present in an actual accelerator, and in particular, we motivate the inclusion of sextupoles to cure chromatic effects. To study the effects of the nonlinear terms, we next discuss canonical perturbation theory which leads us to nonlinear resonances. After showing a few examples of perturbation theory, we abandon it when very close to a resonance. This leads to the study of an isolated resonance in one degree of freedom with a 'time'-dependent Hamiltonian. We see the familiar resonance structure in phase space which is simply closed islands when the nonlinear amplitude dependence of the frequency or 'tune' is included. To show the limits of the validity of the isolated resonance approximation, we discuss two criteria for the onset of chaotic motion. Finally, we study an isolated coupling resonance in two degrees of freedom with a 'time'-dependent Hamiltonian and calculate the two invariants in this case. This leads to a surface of section which is a 2-torus in 4-dimensional phase space. However, we show that it remains a 2-torus when projected into particular 3-dimensional subspaces, and thus can be viewed in perspective.
- Research Organization:
- Stanford Linear Accelerator Center, Menlo Park, CA (USA)
- DOE Contract Number:
- AC03-76SF00515
- OSTI ID:
- 6396968
- Report Number(s):
- SLAC-PUB-3836; CONF-850114-4; ON: DE86005670
- Resource Relation:
- Conference: Joint US/CERN School on Particle Accelerators topical course on non-linear dynamics, Sardinia, Italy, 31 Jan 1985
- Country of Publication:
- United States
- Language:
- English
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BEAM DYNAMICS
NONLINEAR PROBLEMS
CYCLIC ACCELERATORS
RESONANCE
BETATRON OSCILLATIONS
EQUATIONS OF MOTION
HAMILTONIAN FUNCTION
PERTURBATION THEORY
ACCELERATORS
DIFFERENTIAL EQUATIONS
EQUATIONS
FUNCTIONS
OSCILLATIONS
PARTIAL DIFFERENTIAL EQUATIONS
430200* - Particle Accelerators- Beam Dynamics
Field Calculations
& Ion Optics