Nekhoroshev Stability Estimates for Symplectic Maps and Physical Applications
The Nekhoroshev stability estimates for symplectic maps of R 2d are presented. After an illustration of the basic ideas in a simple hamiltonian model, the dynamics of a charged particle in the magnetic lattice of an accelerator is considered and it is shown to be conveniently described by a symplectic map. After recalling the basic properties of the Birkhoff normal forms, we state a general theorem on the stability of the orbits and sketch the proof. The factorial divergence of the Birkhoff series appears to arise from nonlinearity as well as from the divisors, and a functional equation with interesting analytic properties is obtained from the majorant series.
KeywordsNormal Form Diophantine Condition Majorant Series Magnetic Lattice Birkhoff Normal Form
Unable to display preview. Download preview PDF.
- A.N. Kolmogorov, Dokl. Akad. Nauk SSSR, 98, 4, 527–530, (1954). V.l. Arnol’d, Russ. Math. Surv., 18, 5, 9–36, and 6, 85–191 (1963). J. Moser, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. 1, 1, (1962).Google Scholar
- A. Giorgilli and L. Galgani, Cel. Mech., 37, 95–112, (1985). G. Benettin and G. Gallavotti, J. Stat. Phys., 44, 293–338, (1986). A. Giorgilli, Recent developments of perturbation theory of classical hamiltonian systems, in Nonlinear Dynamics Ed. by G. Turchetti, World Scientific, Singapore (1989)MathSciNetADSCrossRefGoogle Scholar
- Lecture Notes in Physics N° 247, Springer Verlag (1986)Google Scholar
- A.J. Liechtenberg, M.A. Liebermann Regular and stochastic motions in dynamical systems, Springer Verlag (1983)Google Scholar
- A. Bazzani, M. Malavasi, S. Siboni and G. Turchetti, Perturbative methods in the analysis of magnetic Gelds structure, Phys. Dep. Univ. of Bologna preprint (1989)Google Scholar
- G. Turchetti, Perturbative Methods for Hamiltonian Maps, in Methods and Applications of Nonlinear Dynamics, A.W. Saenz editor, World Scientific, Singapore, 95–154, (1988). G. Servizi, G. Turchetti, Il Nuovo Cimento B95, 121–154 (1986)Google Scholar
- A. Bazzani, S. Marmi, G. Turchetti Nechoroshev estimates for isochronous symplectic maps Phys. Dep. Univ. of Bologna preprint submitted to Celestial Mechanics.Google Scholar
- D. Bessis, S. Marmi, and G. Turchetti, work in progress.Google Scholar
- M. Malavasi, S. Marmi, G. Servizi and G. Turchetti in preparationGoogle Scholar
- J. Palis and J.C. Yoccoz, Rigidity of Centralizers of Diffeomorphisms, Ann. Ec. Norm. Sup., in press, (1988).Google Scholar