×

Escape to infinity in the presence of magnetic fields. (English) Zbl 1250.78016

The result obtained here is as follows. If all the cyclic wires lie on a plane \(\pi\), then escape to infinity is possible if the initial conditions \((x_0,\dot x_0)\) for the equation \(\ddot x=\dot x\wedge B\) satisfy \(x_0\in\pi\), \(\dot x_0\in\pi\) and \(\| x_0\|\) is large, \(\|\cdot\|\) being the standard Euclidean norm. This result also holds in the relativistic case.

MSC:

78A35 Motion of charged particles
34A34 Nonlinear ordinary differential equations and systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] S. M. Ulam, Problems in modern mathematics, Science Editions John Wiley & Sons, Inc., New York, 1964. · Zbl 0137.24201
[2] J. Sarvas, Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem, Phys. Med. Biol. 32 (1987), 11-22.
[3] F. G. Gascón and D. Peralta-Salas, Motion of a charge in the magnetic field created by wires: impossibility of reaching the wires, Phys. Lett. A 333 (2004), no. 1-2, 72 – 78. · Zbl 1123.78303 · doi:10.1016/j.physleta.2004.09.084
[4] F. G. Gascon and D. Peralta-Salas, Escape to infinity in a Newtonian potential, J. Phys. A 33 (2000), no. 30, 5361 – 5368. · Zbl 0977.34028 · doi:10.1088/0305-4470/33/30/307
[5] F. G. Gascon and D. Peralta-Salas, Escape to infinity under the action of a potential and a constant electromagnetic field, J. Phys. A 36 (2003), no. 23, 6441 – 6455. · Zbl 1094.70509 · doi:10.1088/0305-4470/36/23/310
[6] Y. Matsuno, Two-dimensional dynamical system associated with Abel’s nonlinear differential equation, J. Math. Phys. 33 (1992), no. 1, 412 – 421. · Zbl 0761.34028 · doi:10.1063/1.529923
[7] Alain Goriely and Craig Hyde, Finite-time blow-up in dynamical systems, Phys. Lett. A 250 (1998), no. 4-6, 311 – 318. · Zbl 0946.34074 · doi:10.1016/S0375-9601(98)00822-6
[8] C. Marchioro, Solution of a three-body scattering problem in one dimension, J. Math. Phys. 11 (1970), 2193-2196.
[9] L.P. Fulcher, B.F. Davis, D.A. Rowe, An approximate method for classical scattering problems, Amer. J. Phys. 44 (1976), 956-959.
[10] L. N. Vaserstein, On systems of particles with finite-range and/or repulsive interactions, Comm. Math. Phys. 69 (1979), no. 1, 31 – 56.
[11] G. Galperin, Asymptotic behaviour of particle motion under repulsive forces, Comm. Math. Phys. 84 (1982), no. 4, 547 – 556. · Zbl 0518.70012
[12] Eugene Gutkin, Integrable Hamiltonians with exponential potential, Phys. D 16 (1985), no. 3, 398 – 404. · Zbl 0584.58024 · doi:10.1016/0167-2789(85)90017-X
[13] Eugene Gutkin, Asymptotics of trajectories for cone potentials, Phys. D 17 (1985), no. 2, 235 – 242. · Zbl 0587.58025 · doi:10.1016/0167-2789(85)90008-9
[14] V.J. Menon, D.C. Agrawal, Solar escape revisited, Amer. J. Phys. 54 (1986), 752-753.
[15] Eugene Gutkin, Continuity of scattering data for particles on the line with directed repulsive interactions, J. Math. Phys. 28 (1987), no. 2, 351 – 359. · Zbl 0628.70007 · doi:10.1063/1.527666
[16] Andrea Hubacher, Classical scattering theory in one dimension, Comm. Math. Phys. 123 (1989), no. 3, 353 – 375. · Zbl 0675.34016
[17] Vinicio Moauro, Piero Negrini, and Waldyr Muniz Oliva, Analytic integrability for a class of cone potential Hamiltonian systems, J. Differential Equations 90 (1991), no. 1, 61 – 70. · Zbl 0721.34010 · doi:10.1016/0022-0396(91)90161-2
[18] G. Fusco and W. M. Oliva, Integrability of a system of \? electrons subjected to Coulombian interactions, J. Differential Equations 135 (1997), no. 1, 16 – 40. · Zbl 0878.58051 · doi:10.1006/jdeq.1996.3171
[19] Courtney S. Coleman, Boundedness and unboundedness in polynomial differential systems, Nonlinear Anal. 8 (1984), no. 11, 1287 – 1294. · Zbl 0512.34030 · doi:10.1016/0362-546X(84)90016-6
[20] Zhi Fen Zhang, Tong Ren Ding, Wen Zao Huang, and Zhen Xi Dong, Qualitative theory of differential equations, Translations of Mathematical Monographs, vol. 101, American Mathematical Society, Providence, RI, 1992. Translated from the Chinese by Anthony Wing Kwok Leung.
[21] Helmut Röhrl and Sebastian Walcher, Projections of polynomial vector fields and the Poincaré sphere, J. Differential Equations 139 (1997), no. 1, 22 – 40. · Zbl 0885.34028 · doi:10.1006/jdeq.1997.3298
[22] A. Garcia, E. Pérez-Chavela, and A. Susin, A generalization of the Poincaré compactification, Arch. Ration. Mech. Anal. 179 (2006), no. 2, 285 – 302. · Zbl 1085.58006 · doi:10.1007/s00205-005-0389-y
[23] George Arfken, Mathematical methods for physicists, Academic Press, New York-London, 1966. · Zbl 0135.42304
[24] Manfredo P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976. Translated from the Portuguese. · Zbl 0326.53001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.