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Stability and instability for particle dynamics in magnetic fields

  • Alberto Enciso
  • Daniel Peralta-Salas
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Abstract

We study zero-speed equilibria of the Newton-Lorentz equation ẍ=ẋ× B(x) in R3, which describes the motion of a charged particle in a magnetic field B. We prove Lyapunov instability wherever the field B does not vanish (under C1 regularity), and at every isolated zero of B provided it is real-analytic. These results do not assume that the field is divergence-free. In contrast, we construct a compactly supported divergence-free C∞ field that is flat at an isolated zero and makes the corresponding equilibrium Lyapunov stable.

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Cite

BibTeXRIS
BibTeX
@misc{enciso2026stability,
  title = {{Stability and instability for particle dynamics in magnetic fields}},
  author = {Alberto Enciso and Daniel Peralta-Salas},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27893},
  eprint = {2610.27893},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27893},
}
RIS
TY  - UNPB
TI  - Stability and instability for particle dynamics in magnetic fields
AU  - Alberto Enciso
AU  - Daniel Peralta-Salas
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.27893
UR  - https://arxiv.org/abs/2610.27893
AB  - We study zero-speed equilibria of the Newton-Lorentz equation ẍ=ẋ× B(x) in R^3, which describes the motion of a charged particle in a magnetic field B. We prove Lyapunov instability wherever the field B does not vanish (under C^1 regularity), and at every isolated zero of B provided it is real-analytic. These results do not assume that the field is divergence-free. In contrast, we construct a compactly supported divergence-free C^∞ field that is flat at an isolated zero and makes the corresponding equilibrium Lyapunov stable.
ER  -