Escape to infinity in the presence of magnetic fields
Authors:
A. Díaz-Cano and F. González-Gascón
Journal:
Quart. Appl. Math. 70 (2012), 45-51
MSC (2010):
Primary 78A35; Secondary 34A34
DOI:
https://doi.org/10.1090/S0033-569X-2011-01248-4
Published electronically:
August 26, 2011
MathSciNet review:
2920614
Full-text PDF Free Access
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Additional Information
Abstract: Escape to infinity is proved to occur when a charge moves under the action of the magnetic field created by a finite number of planar closed wires.
References
- S. M. Ulam, Problems in modern mathematics, Science Editions John Wiley & Sons, Inc., New York, 1964. MR 0280310
- J. Sarvas, Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem, Phys. Med. Biol. 32 (1987), 11–22.
- 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. MR 2101963, DOI https://doi.org/10.1016/j.physleta.2004.09.084
- F. G. Gascon and D. Peralta-Salas, Escape to infinity in a Newtonian potential, J. Phys. A 33 (2000), no. 30, 5361–5368. MR 1783734, DOI https://doi.org/10.1088/0305-4470/33/30/307
- 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. MR 1987441, DOI https://doi.org/10.1088/0305-4470/36/23/310
- Y. Matsuno, Two-dimensional dynamical system associated with Abel’s nonlinear differential equation, J. Math. Phys. 33 (1992), no. 1, 412–421. MR 1141541, DOI https://doi.org/10.1063/1.529923
- Alain Goriely and Craig Hyde, Finite-time blow-up in dynamical systems, Phys. Lett. A 250 (1998), no. 4-6, 311–318. MR 1666111, DOI https://doi.org/10.1016/S0375-9601%2898%2900822-6
- C. Marchioro, Solution of a three-body scattering problem in one dimension, J. Math. Phys. 11 (1970), 2193-2196.
- L.P. Fulcher, B.F. Davis, D.A. Rowe, An approximate method for classical scattering problems, Amer. J. Phys. 44 (1976), 956–959.
- L. N. Vaserstein, On systems of particles with finite-range and/or repulsive interactions, Comm. Math. Phys. 69 (1979), no. 1, 31–56. MR 547525
- G. Galperin, Asymptotic behaviour of particle motion under repulsive forces, Comm. Math. Phys. 84 (1982), no. 4, 547–556. MR 667760
- Eugene Gutkin, Integrable Hamiltonians with exponential potential, Phys. D 16 (1985), no. 3, 398–404. MR 805712, DOI https://doi.org/10.1016/0167-2789%2885%2990017-X
- Eugene Gutkin, Asymptotics of trajectories for cone potentials, Phys. D 17 (1985), no. 2, 235–242. MR 815287, DOI https://doi.org/10.1016/0167-2789%2885%2990008-9
- V.J. Menon, D.C. Agrawal, Solar escape revisited, Amer. J. Phys. 54 (1986), 752–753.
- Eugene Gutkin, Continuity of scattering data for particles on the line with directed repulsive interactions, J. Math. Phys. 28 (1987), no. 2, 351–359. MR 872012, DOI https://doi.org/10.1063/1.527666
- Andrea Hubacher, Classical scattering theory in one dimension, Comm. Math. Phys. 123 (1989), no. 3, 353–375. MR 1003425
- 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. MR 1094449, DOI https://doi.org/10.1016/0022-0396%2891%2990161-2
- G. Fusco and W. M. Oliva, Integrability of a system of $N$ electrons subjected to Coulombian interactions, J. Differential Equations 135 (1997), no. 1, 16–40. MR 1434913, DOI https://doi.org/10.1006/jdeq.1996.3171
- Courtney S. Coleman, Boundedness and unboundedness in polynomial differential systems, Nonlinear Anal. 8 (1984), no. 11, 1287–1294. MR 764913, DOI https://doi.org/10.1016/0362-546X%2884%2990016-6
- 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. MR 1175631
- Helmut Röhrl and Sebastian Walcher, Projections of polynomial vector fields and the Poincaré sphere, J. Differential Equations 139 (1997), no. 1, 22–40. MR 1467351, DOI https://doi.org/10.1006/jdeq.1997.3298
- A. Garcia, E. Pérez-Chavela, and A. Susin, A generalization of the Poincaré compactification, Arch. Ration. Mech. Anal. 179 (2006), no. 2, 285–302. MR 2209132, DOI https://doi.org/10.1007/s00205-005-0389-y
- George Arfken, Mathematical methods for physicists, Academic Press, New York-London, 1966. MR 0205512
- Manfredo P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976. Translated from the Portuguese. MR 0394451
References
- S. Ulam, Problems in modern mathematics, Science Editions, John Wiley & Sons, Inc., 1964. MR 0280310 (43:6031)
- J. Sarvas, Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem, Phys. Med. Biol. 32 (1987), 11–22.
- F. González-Gascón, D. Peralta-Salas, Motion of a charge in the magnetic field created by wires: impossibility of reaching the wires, Phys. Lett. A. 333 (2004), 72–78. MR 2101963 (2005g:78009)
- F. González-Gascón, D. Peralta-Salas, Escape to infinity in a Newtonian potential, J. Phys. A 33 (2000), 5361–5368. MR 1783734 (2001j:70009)
- F. González-Gascón, D. Peralta-Salas, Escape to infinity under the action of a potential and a constant electromagnetic field, J. Phys. A 36 (2003), 6441–6455. MR 1987441 (2004f:70042)
- Y. Matsuno, Two-dimensional dynamical system associated with Abel’s nonlinear differential equation, J. Math. Phys. 33 (1992), 412–421. MR 1141541 (92h:34025)
- A. Goriely, C. Hyde, Finite-time blow-up in dynamical systems, Phys. Lett. A 250 (1998), 311–318. MR 1666111 (2000e:34056)
- C. Marchioro, Solution of a three-body scattering problem in one dimension, J. Math. Phys. 11 (1970), 2193-2196.
- L.P. Fulcher, B.F. Davis, D.A. Rowe, An approximate method for classical scattering problems, Amer. J. Phys. 44 (1976), 956–959.
- L. Vaserstein, On systems of particles with finite-range and/or repulsive interactions, Commun. Math. Phys. 69 (1979), 31-56. MR 547525 (81b:58028)
- G. Galperin, Asymptotic behaviour of particle motion under repulsive forces, Commun. Math. Phys. 84 (1982), 547-556. MR 667760 (83m:70023)
- E. Gutkin, Integrable Hamiltonians with exponential potential, Phys. D 16 (1985), 398–404. MR 805712 (86k:58036)
- E. Gutkin, Asymptotics of trajectories for cone potentials, Phys. D 17 (1985), 235–242. MR 815287 (87c:58040)
- V.J. Menon, D.C. Agrawal, Solar escape revisited, Amer. J. Phys. 54 (1986), 752–753.
- E. Gutkin, Continuity of scattering data for particles on the line with directed repulsive interactions, J. Math. Phys. 28 (1987), 351–359. MR 872012 (88f:70011)
- A. Hubacher, Classical scattering theory in one dimension, Comm. Math. Phys. 123 (1989), 353–375. MR 1003425 (90f:58062)
- V. Moauro, P. Negrini, W.M. Oliva, Analytic integrability for a class of cone potential Hamiltonian systems, J. Differential Equations 90 (1991), 61–70. MR 1094449 (92b:58072)
- G. Fusco, W.M. Oliva, Integrability of a system of $N$ electrons subjected to Coulombian interactions, J. Differential Equations 135 (1997), 16–40. MR 1434913 (98c:70012)
- C. Coleman, Boundedness and unboundedness in polynomial differential systems, Nonlinear Anal. 8 (1984), 1287–1294. MR 764913 (86b:34064)
- Z.F. Zhang, T.R. Ding, W.Z. Huang, Z.X. Dong, Qualitative theory of differential equations, Translations of Mathematical Monographs 101, American Mathematical Society, 1992. MR 1175631 (93h:34002)
- H. Röhrl, S. Walcher, Projections of polynomial vector fields and the Poincaré sphere, J. Differential Equations 139 (1997), 22–40. MR 1467351 (98j:34050)
- A. García, E. Pérez-Chavela, A. Susin, A generalization of the Poincaré compactification, Arch. Ration. Mech. Anal. 179 (2006), 285–302. MR 2209132 (2007f:37026)
- G. Arfken, Mathematical methods for physicists, Academic Press, 1966. MR 0205512 (34:5339)
- M.P. do Carmo, Differential Geometry of curves and surfaces, Prentice-Hall, 1976. MR 0394451 (52:15253)
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Additional Information
A. Díaz-Cano
Affiliation:
IMI and Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, E-28040 Madrid, Spain
Email:
adiaz@mat.ucm.es
F. González-Gascón
Affiliation:
Departamento de Física Teórica II, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain
Email:
fggascon@fis.ucm.es
Keywords:
Escape to infinity,
magnetic field,
Lorentz equation
Received by editor(s):
April 30, 2010
Published electronically:
August 26, 2011
Additional Notes:
The first author was partially supported by Spanish GAAR MTM2008-00272 and GAAR Grupos UCM 910444
Article copyright:
© Copyright 2011
Brown University