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### Interaction of two charges in a uniform magnetic field : II. spatial problem

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Pinheiro, D. and MacKay, R. S. (Robert Sinclair).
(2008)
*Interaction of two charges in a uniform magnetic field : II. spatial problem.*
Journal of Nonlinear Science, Vol.18
(No.6).
pp. 615-666.
ISSN 0938-8974

**Full text not available from this repository.**

Official URL: http://dx.doi.org/10.1007/s00332-008-9022-1

## Abstract

The interaction of two charges moving in R-3 in a magnetic field B can be formulated as a Hamiltonian system with six degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotation symmetry, we reduce this system to one with three degrees of freedom. For special values of the conserved quantities, choices of parameters or restriction to the coplanar case, we obtain systems with two degrees of freedom. Specialising to the case of Coulomb interaction, these reductions enable us to obtain many qualitative features of the dynamics. For charges of the same sign, the gyrohelices either "bounce-back", "pass-through", or exceptionally converge to coplanar solutions. For charges of opposite signs, we decompose the state space into "free" and "trapped" parts with transitions only when the particles are coplanar. A scattering map is defined for those trajectories that come from and go to infinite separation along the field direction. It determines the asymptotic parallel velocities, guiding centre field lines, magnetic moments and gyrophases for large positive time from those for large negative time. In regimes where gyrophase averaging is appropriate, the scattering map has a simple form, conserving the magnetic moments and parallel kinetic energies (in a frame moving along the field with the centre of mass) and rotating or translating the guiding centre field lines. When the gyrofrequencies are in low-order resonance, however, gyrophase averaging is not justified and transfer of perpendicular kinetic energy is shown to occur. In the extreme case of equal gyrofrequencies, an additional integral helps us to analyse further and prove that there is typically also transfer between perpendicular and parallel kinetic energy.

[error in script] [error in script]Item Type: | Journal Article |
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Subjects: | Q Science > QA Mathematics Q Science > QC Physics |

Divisions: | Faculty of Science > Mathematics |

Library of Congress Subject Headings (LCSH): | Hamiltonian systems, Euclidean algorithm, Reconstruction (Graph theory), Magnetic fields |

Journal or Publication Title: | Journal of Nonlinear Science |

Publisher: | Springer |

ISSN: | 0938-8974 |

Date: | December 2008 |

Volume: | Vol.18 |

Number: | No.6 |

Number of Pages: | 52 |

Page Range: | pp. 615-666 |

Identification Number: | 10.1007/s00332-008-9022-1 |

Status: | Peer Reviewed |

Publication Status: | Published |

Access rights to Published version: | Restricted or Subscription Access |

Funder: | Fundação para a Ciência e a Tecnologia (FCT), Universidade do Porto. Centro de Matematica |

Grant number: | SFRH/BD/9239/2002 (FCT), SFRH/BPD/27151/2006 (FCT) |

References: | Abraham, R., Marsden, J.: Foundations of Mechanics, 2nd edn. Benjamin/Cummings, Reading (1978) Anderegg, F., Huang, X.P., Driscoll, C.F., Hollmann, E.M., O’Neil, T.M., Dubin, D.H.E.: Test particle transport due to long range interactions. Phys. Rev. Lett. 78, 2128–2131 (1997) Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989) Avron, J.E., Herbst, I.W., Simon, B.: Separation of center of mass in homogeneous magnetic fields. Ann. Phys. 114, 431–451 (1978) Balescu, R.: Transport Processes in Plasma, vol. I. North-Holland, Amsterdam (1988) Cushman, R., Bates, L.: Global Aspects of Classical Integrable Systems. Birkhäuser, Basel (1997) Driscoll, C.F., Anderegg, F., Dubin, D.H.E., Jin, D.Z., Kriesel, J.M., Hollmann, E.M., O’Neil, T.M.: Shear reduction of collisional transport: experiments and theory. Phys. Plasmas 9, 1905–1914 (2002) Dubin, D.H.E.: Collisional transport in non-neutral plasmas. Phys. Plasmas 5, 1688–1694 (1998) Efstathiou, K., Cushman, R.H., Sadovskií, D.A.: Hamiltonian Hopf bifurcation of the hydrogen atom in crossed fields. Physica D 194, 250–274 (2004) Grotta Raggazo, C., Koiller, J., Oliva,W.M.: On the motion of two-dimensional vortices with mass. J. Nonlinear Sci. 4, 375–418 (1994) Gutzwiller, M.C.: Chaos in Classical and Quantum Mechanics. Interdisciplinary Applied Mathematics. Springer, Berlin (1990) Helander, P., Sigmar, A.J.: Collisional Transport in Magnetized Plasmas. Cambridge University Press, Cambridge (2002) Littlejohn, R.G.: A guiding center Hamiltonian: a new approach. J. Math. Phys. B 20, 2445–2458 (1979) Lochak, P., Meunier, C.: Mutiphase Averaging for Classical Systems: With Applications to Adiabatic Theorems. Applied Mathematical Sciences, vol. 72. Springer, New York (1988) MacKay, R.S.: Flux over a saddle. Phys. Lett. A 145(8,9), 425–427 (1990) Marsden, J., Ratiu, T.: Introduction to Mechanics and Symmetry. Springer, New York (1999) Ott, E., Tél, T.: Chaotic scattering: an introduction. Chaos 3(4), 417–426 (1993) Pinheiro, D.: Interaction of two charges in a uniform magnetic field. Ph.D. thesis, University of Warwick (2006) Pinheiro, D., MacKay, R.S.: Interaction of two charges in a uniform magnetic field: I. Planar problem. Nonlinearity 19(8), 1713–1745 (2006) Psimopoulos, M., Li, D.: Cross field thermal transport in highly magnetized plamas. Proc. R. Soc. Lond. 437, 55–65 (1992) Schmelcher, P., Cederbaum, L.S.: Regularity and chaos in the center of mass motion of the hydrogen atom in a magnetic field. Z. Phys. D 24(4), 311–323 (1992) Schmelcher, P., Cederbaum, L.S.: Intermittent chaos in Hamiltonian systems: The three-dimensional hydrogen atom in magnetic fields. Phys. Rev. A 47(4), 2634–2639 (1993) Tannenwald, L.M.: Coulomb scattering in a very strong magnetic field. Phys. Rev. 113(6), 1396–1405 (1959) Tanner, G., Hansen, K.T., Main, J.: The semiclassical resonance spectrum of hydrogen in a constant magnetic field. Nonlinearity 9, 1641–1670 (1996) |

URI: | http://wrap.warwick.ac.uk/id/eprint/28908 |

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