Abstract
We will compare two different techniques to solve a problem of motion of a charged particle inside magnetic dipole field. One “classical” and the other using pedal coordinates. We will show that even though the classical approach gives an exact solution in terms of known function, pedal coordinates offer much better understanding of the solution and also offer a mean to manipulate the obtained orbits in order to be able to link them with existing curves and other force problems.
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The author was supported by the GAČR grant no. 21-27941S and RVO funding 47813059.
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Appendix: Gallery
Appendix: Gallery
Orbits of (4) corresponding to the cases from left to right: Δ = 0.00394, Δ = 0.04456,Δ = 0.00040, respectively. Bounded and unbounded components are both present
Unbounded orbits of (4) corresponding to the cases from left to right: Δ = −0.0663, Δ = −3.5297, Δ = −0.1546, respectively. There are always two unbounded components
Left: Orbit of (4) in the case Δ = 0. Right: Its dual curve. The dual curve is a portion of the same curve but rotated 90∘
Trajectory of solution of (4) together with its square root and its dual curve
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Blaschke, P. (2023). Pedal Coordinates and Orbits Inside Magnetic Dipole Field. In: Kielanowski, P., Dobrogowska, A., Goldin, G.A., Goliński, T. (eds) Geometric Methods in Physics XXXIX. WGMP 2022. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-30284-8_14
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DOI: https://doi.org/10.1007/978-3-031-30284-8_14
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