Motions of a charged particle in the electromagnetic field induced by a non-stationary current. (English) Zbl 1483.78001
Summary: In this paper we study the non-relativistic dynamic of a charged particle in the electromagnetic field induced by a periodically time dependent current \(J\) along an infinitely long and infinitely thin straight wire. The motions are described by the Lorentz-Newton equation, in which the electromagnetic field is obtained by solving the Maxwell’s equations with the current distribution \(\vec{J}\) as data. We prove that many features of the integrable time independent case are preserved. More precisely, introducing cylindrical coordinates, we prove the existence of (non-resonant) radially periodic motions that are also of twist type. In particular, these solutions are Lyapunov stable and accumulated by subharmonic and quasiperiodic motions.
MSC:
| 78A35 | Motion of charged particles |
| 78A25 | Electromagnetic theory (general) |
| 35B10 | Periodic solutions to PDEs |
| 35B35 | Stability in context of PDEs |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
Keywords:
Lorentz force; singular potentials; Maxwell’s equations; non-steady current; periodic solutions of twist type; stabilityReferences:
| [1] | Griffiths, D. J., Introduction to Electrodynamics (1962), Prentice Hall: Prentice Hall New Jersey |
| [2] | Jackson, J. D., Classical Electrodynamics (1999), John Wiley & Sons · Zbl 0920.00012 |
| [3] | Planck, M., Das prinzip der relativität und die grundgleichungen der mechanik, Verh. Dtsch. Phys. Ges., 4, 136-141 (1906) · JFM 37.0721.04 |
| [4] | Poincaré, H., Sur la dynamique de l’électron, Rend. Circ. Mat. Palermo, 21, 129-176 (1906) · JFM 37.0886.01 |
| [5] | Aguirre, J.; Luque, A.; Peralta-Salas, D., Motion of charged particles in magnetic fields created by symmetric configurations of wires, Physica D, 239, 654-674 (2010) · Zbl 1186.37103 |
| [6] | Gascón, F. G.; Peralta-Salas, D., Motion of a charge in the magnetic field created by wires: impossibility of reaching the wires, Phys. Lett. A, 333, 72-78 (2004) · Zbl 1123.78303 |
| [7] | Gascón, F. G.; Peralta-Salas, D., Some properties of the magnetic fields generated by symmetric configurations of wires, Physica D, 206, 109-120 (2005) · Zbl 1066.78004 |
| [8] | Arcoya, D.; Bereanu, C.; Torres, P. J., Critical point theory for the Lorentz force equation, Arch. Ration. Mech. Anal., 232, 1685-1724 (2019) · Zbl 1421.35095 |
| [9] | Arcoya, D.; Bereanu, C.; Torres, P. J., Lusternik-Schnirelmann theory for the action integral of the Lorentz force equation, Calc. Var. Partial Differential Equations, 59, 2, 50 (2020) · Zbl 1434.78006 |
| [10] | Garzón, M.; Torres, P. J., Periodic solutions for the Lorentz force equation with singular potentials, Nonlinear Anal. RWA, 56, Article 103162 pp. (2020) · Zbl 1456.34041 |
| [11] | Hau, L. V.; Burns, M. M.; Golovchenko, J. A., Bound states of guided matter waves: An atom and a charged wire, Phys. Rev. A, 45, 6468-6478 (1992) |
| [12] | King, C.; Leśniewski, A., Periodic motion of atoms near a charged wire, Lett. Math. Phys., 39, 367-378 (1997) · Zbl 0869.34037 |
| [13] | Lei, J.; Zhang, M., Twist property of periodic motion of an atom near a charged wire, Lett. Math. Phys., 60, 9-17 (2002) · Zbl 1002.78006 |
| [14] | Ortega, R., Periodic solutions of a Newtonian equation: Stability by the third approximation, J. Differential Equations, 128, 491-518 (1996) · Zbl 0855.34058 |
| [15] | Torres, P. J.; Zhang, M., Twist periodic solutions of repulsive singular equations, Nonlinear Anal., 56, 591-599 (2004) · Zbl 1058.34052 |
| [16] | Siegel, C.; Moser, J., Lectures on Celestial Mechanics (1971), Springer-Verlag · Zbl 0312.70017 |
| [17] | Marò, S., Relativistic pendulum and invariant curves, Discrete Contin. Dyn. Syst., 35, 1139-1162 (2015) · Zbl 1302.70050 |
| [18] | Marò, S.; Ortega, V., Twist dynamics and Aubry-Mather sets around a periodically perturbed point-vortex, Differential Equations, 269, 3624-3651 (2020) · Zbl 1448.37070 |
| [19] | Ortega, R., Asymmetric oscillators and twist mappings, J. Lond. Math. Soc., 53, 325-342 (1996) · Zbl 0860.34017 |
| [20] | Ortega, R., Twist mappings, invariant curves and periodic differential equations, (Nonlinear Analysis and its Applications to Differential Equations (Lisbon, 1998). Nonlinear Analysis and its Applications to Differential Equations (Lisbon, 1998), Progr. Nonlinear Differential Equations Appl., vol. 43 (2001), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 85-112 · Zbl 1172.34326 |
| [21] | Coddington, R.; Levinson, N., Theory of Ordinary Differential Equations (1955), Tata McGraw-Hill Education · Zbl 0064.33002 |
| [22] | Hartman, P., Ordinary Differential Equations (2002), SIAM · Zbl 0125.32102 |
| [23] | Mather, J. N., Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology, 21, 457-467 (1982) · Zbl 0506.58032 |
| [24] | Meyer, K. R.; Hall, G. R., Introduction to Hamiltonian Dynamical Systems and the N-Body Problem (1992), Springer-Verlag: Springer-Verlag New York · Zbl 0743.70006 |
| [25] | Moser, J., Recent developments in the theory of Hamiltonian systems, SIAM Rev., 28, 459-485 (1986) · Zbl 0606.58022 |
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.
