Recent Applications of Hamiltonian Dynamics to Accelerator Physics
The use of superconducting magnets in large hadron accelerators has raised new interest in the nonlinear effects in accelerator physics (Scandale and Turchetti, 1990): indeed, both analytical tools and reliable numerical methods are required to design future accelerators. The standard numerical approach is based on the concept of symplectic transfer maps (Iselin and Niederer, 1988; Schmidt, 1990); the corresponding analytical tool is the theory of normal forms (Turchetti, 1988; Bazzani et al., 1993a), which is the natural generalization of canonical perturbation theory for flows to transfer maps. Normal forms for symplectic maps have the big advantage of being easily implement able in computer codes (Servizi and Turchetti, 1984), allowing the automatic computation of high perturbative orders; moreover, even if the series are generically divergent such as in the hamiltonian case, a detailed analysis of the mechanism of divergence was carried out, allowing to use the approximation provided by truncated normal forms in judiciously chosen domains (Bazzani et al., 1993a and 1993b).
KeywordsNormal Form Stability Domain Nonlinear Motion Magnetic Lattice Accelerator Physic
Unable to display preview. Download preview PDF.
- Arnold, V. I., and Avez, A., 1968, “Ergodic problems of classical mechanics,” V. A. Benjamin, New York.Google Scholar
- Bazzani, A., and Turchetti, G., 1991, Polyn: a program for algorithmic manipulations of polynomials, CBRN SL report 91-13 AP.Google Scholar
- Bazzani, A., Servizi, G., Todesco, E., and Turchetti, G., 1993a, A normal form approach to the theory of nonlinear betatronic motion, submitted as CERN yellow report.Google Scholar
- Giovannozzi, M., 1992, Analysis of the Stability Domain for the Hénon Map, CERN SL report 92-23 AP.Google Scholar
- Giovannozzi, M., and Schmidt, F., 1993, General normal form procedure to correct tune-shift and non-linear chromaticity for large accelerators like the LHC, in: “1993 Particle Accelerator Conference”, to be published.Google Scholar
- Iselin, F.C., and Niederer, J., 1988, The MAD program 7.2 User reference manual, CERN LEP report 88-38 TH.Google Scholar
- Scandale, W., 1989, Status report on the design of the LHC lattice, CERN LHC Note 68.Google Scholar
- Scandale, W., and Turchetti, G., 1990, “Nonlinear problems in future particle accelerators,” World Scientific, Singapore.Google Scholar
- Scandale, W., Schmidt, F., and Todesco, E., 1992, Compensation of the tuneshift in the LHC, using normal form techniques, Part. Accel. 35:53–88.Google Scholar
- Schmidt, F., 1990, SIXTRACK: Single particle tracking code treating transverse motion with synchrotron oscillations in a symplectic manner, CERN SL report 90-11 AP.Google Scholar
- Steffen, K., 1985, Basic course in accelerator optics, CERN 85-19:25–63.Google Scholar
- Turchetti, G., 1988, Perturbative methods for hamiltonian maps, in: “Methods and applications of nonlinear dynamics,” A. W. Saenz, ed., World Scientific, Singapore.Google Scholar