Perturbed ion traps: A generalization of the three-dimensional Hénon-Heiles problem

Chaos

Volumn 12, Issue 1, 2002, Pages 87-99

  Lanchares, V ^a   Pascual, A I ^a   Palacian J ^b   Yanguas, P ^b   Salas, J P ^a  

^a UNIVERSITY OF LA RIOJA   (Spain)

^b PUBLIC UNIVERSITY OF NAVARRE   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036489720     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1449957     Document Type: Article

References (30)

1. G. Contopoulos, "A third integral of motion in a galaxy," Z. Astrophys. 49, 273-291 (1960).
2. V. Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies (Academic, New York, 1967).
3. G. Contopoulos, in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, edited by G. Casati and J. Ford, Lecture Notes in Physics, Vol. 93 (Springer, Berlin, 1979).
4. G.M. Zaslavski, R.Z. Sagdeev, D.A. Usikov, A.A. Chernikov, and A.R. Sagdeeva, Weak Chaos and Quasi-Regular Patterns, Cambridge Nonlinear Science Series, Vol. 1 (Cambridge University Press, New York, 1991).
5. R. Dilão and R. Alves-Pires, Nonlinear Dynamics in Particle Accelerators, World Scientific Series on Nonlinear Science Series A, Vol. 23 (World Scientific, Singapore, 1996).
6. H. Friedrich and D. Wintgen, "The hydrogen atom in a uniform magnetic field: An example of chaos," Phys. Rep. 183, 37-79 (1989).
7. M. Gutzwiller, Chaos in Classical and Quantum Mechanics, Interdisciplinary Applied Mathematics, Vol. 1 (Springer, New York, 1990).
8. F.M. Penning, "Introduction of an axial magnetic field in the discharge between two coaxial cylinders," Physica (Amsterdam) 3, 873-894 (1936).
9. H. Dehmelt, "Experiments with an isolated subatomic particle," Rev. Mod. Phys. 62, 525-530 (1990).
10. P.K. Ghosh, Ion Traps, International Series of Monographs on Physics (Oxford University Press, London, 1995).
11. M. Kretzschmar, "Single particle motion in Penning trap: Description in the classical canonical formalism," Phys. Scr. 46, 544-554 (1992).
12. G. Zs. K. Horvath, J.-L. Hernández-Pozos, K. Dholakai, J. Rink, D.M. Segal, and R.C. Thompsom, "Ion dynamics in perturbed quadrupole ion traps," Phys. Rev. A 57, 1944-1956 (1998).
13. M. Hénon and C. Heiles, "The applicability of the third integral of motion: Some numerical experiments," Astron. J. 69, 73-79 (1964).
14. A. Deprit, "Canonical transformations depending on a small parameter," Celest. Mech. 1, 12-30 (1969).
15. J.D. Jackson, Classical Electrodynamics (Wiley, New York, 1998).
16. P. Yanguas, "Integrability, Normalization and Symmetries of Hamiltonian Systems in 1-1-1 Resonance," Ph.D. thesis, Universidad Pública de Navarra, Pamptona, Spain, 1998.
17. S. Ferrer, H. Hanßmann, J. Palacián, and P. Yanguas, "On perturbed oscillators in 1-1-1 resonance: The case of axially symmetric cubic potentials," J. Geom. Phys. 40, 320-369 (2001).
18. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems (Springer, Berlin, 1996).
19. R. Cushman, S. Ferrer, and H. Hanßmann, "Singular reduction of axially symmetric perturbations of the isotropic harmonic oscillator," Nonlinearity 12, 389-410 (1999).
20. S. Ferrer and J. Gárate, "On perturbed 3D elliptic oscillators: A case of critical inclination in galactic dynamics," in New Trends for Hamiltonian Systems and Celestial Mechanics, edited by E.A. Lacomba and J. Llibre, Advanced Series in Nonlinear Dynamics, Vol. 8 (World Scientific, Singapore, 1996), pp. 179-197.
21. S. Ferrer, J. Palacián, and P. Yanguas, "Hamiltonian oscillators in 1-1-1 resonance: Normalization and integrability;" J. Nonlinear Sci. 10, 145-174 (2000).
22. J. Milnor, Morse Theory, Annals of Mathematics Studies (Princeton University Press, Princeton, NJ, 1963), Vol. 51.
23. P.A. Firbi and C.F. Gardiner, Surface Topology (Horwood, New York, 1991).
24. I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Mathematics: Theory & Application Series (Birkhäuser, Boston, 1996).
25. K.R. Meyer, "Generic bifurcation of periodic points," Transp. Res., Part A 149, 95-107 (1970).
26. R. Abraham and J.E. Marsden, Foundations of Mechanics (Dover, Redwood City, CA, 1985).
27. L. Healy and E. Deprit, "Paint by number: Uncovering phase flows of an integrable dynamical system," Comput. Phys. 5, 491-496 (1991).
28. TRANSFORM (Former Research LLC, Sterling, VA 20164, 1996).
29. F. Fassò and G. Benettin, "Composition of Lie transforms with rigorous estimates and applications to Hamiltonian perturbation theory," ZAMP 40, 307-329 (1989).
30. M. Moore and R. Blumel, "Quantum manifestations of order and chaos in the Paul Trap," Phys. Rev. A 48, 3082-3090 (1993).