dorsal/arxiv
View SchemaFailure of geometric electromagnetism in the adiabatic vector Kepler problem
| Authors | J. R. Anglin, J. Schmiedmayer |
|---|---|
| Categories | |
| ArXiv ID | physics/0211062 |
| URL | https://arxiv.org/abs/physics/0211062 |
| DOI | 10.1103/PhysRevA.69.022111 |
| Journal | Phys.Rev.A 69, 022111 (2004) |
Abstract
The magnetic moment of a particle orbiting a straight current-carrying wire may precess rapidly enough in the wire's magnetic field to justify an adiabatic approximation, eliminating the rapid time dependence of the magnetic moment and leaving only the particle position as a slow degree of freedom. To zeroth order in the adiabatic expansion, the orbits of the particle in the plane perpendicular to the wire are Keplerian ellipses. Higher order post-adiabatic corrections make the orbits precess, but recent analysis of this `vector Kepler problem' has shown that the effective Hamiltonian incorporating a post-adiabatic scalar potential (`geometric electromagnetism') fails to predict the precession correctly, while a heuristic alternative succeeds. In this paper we resolve the apparent failure of the post-adiabatic approximation, by pointing out that the correct second-order analysis produces a third Hamiltonian, in which geometric electromagnetism is supplemented by a tensor potential. The heuristic Hamiltonian of Schmiedmayer and Scrinzi is then shown to be a canonical transformation of the correct adiabatic Hamiltonian, to second order. The transformation has the important advantage of removing a $1/r^3$ singularity which is an artifact of the adiabatic approximation.
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"abstract": "The magnetic moment of a particle orbiting a straight current-carrying wire\nmay precess rapidly enough in the wire\u0027s magnetic field to justify an adiabatic\napproximation, eliminating the rapid time dependence of the magnetic moment and\nleaving only the particle position as a slow degree of freedom. To zeroth order\nin the adiabatic expansion, the orbits of the particle in the plane\nperpendicular to the wire are Keplerian ellipses. Higher order post-adiabatic\ncorrections make the orbits precess, but recent analysis of this `vector Kepler\nproblem\u0027 has shown that the effective Hamiltonian incorporating a\npost-adiabatic scalar potential (`geometric electromagnetism\u0027) fails to predict\nthe precession correctly, while a heuristic alternative succeeds. In this paper\nwe resolve the apparent failure of the post-adiabatic approximation, by\npointing out that the correct second-order analysis produces a third\nHamiltonian, in which geometric electromagnetism is supplemented by a tensor\npotential. The heuristic Hamiltonian of Schmiedmayer and Scrinzi is then shown\nto be a canonical transformation of the correct adiabatic Hamiltonian, to\nsecond order. The transformation has the important advantage of removing a\n$1/r^3$ singularity which is an artifact of the adiabatic approximation.",
"arxiv_id": "physics/0211062",
"authors": [
"J. R. Anglin",
"J. Schmiedmayer"
],
"categories": [
"physics.class-ph",
"physics.gen-ph"
],
"doi": "10.1103/PhysRevA.69.022111",
"journal_ref": "Phys.Rev.A 69, 022111 (2004)",
"title": "Failure of geometric electromagnetism in the adiabatic vector Kepler problem",
"url": "https://arxiv.org/abs/physics/0211062"
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