XXXII Workshop on Geometric Methods in Physics 30.06-6.07.2013

Ivailo Mladenov

Planar Motion of Charged Particles in a Magnetic Dipole Field

We consider the system of Newton-Lorentz equations describing the planar non-relativistic motion of a charged particle of unit mass in a transverse magnetic dipole field.
Actually, this system belongs to the class of dynamical systems of two degrees
of freedom whose integrability in the Liouville-Arnold sense is studied in [1].
The magnitude of magnetic dipole field depends only on the distance from the
origin and therefore (see [2]), the corresponding Newton-Lorentz system is
integrable by quadratures since it possesses two functionally independent
integrals of motion. In the present work, the techniques developed by the authors
in [1, 2] are used to represent the trajectories of the particles in explicit
analytic form in terms of Jacobian elliptic functions.  

[1] Vassilev V., Djondjorov P. and Mladenov I.,
Integrable Dynamical Systems of the Frenet–Seret Type.
In: Proc. 9th International Workshop on Complex Structures,
Integrability and Vector Fields, World Scientific,
Singapore 2009, pp. 234-245.
[2] Mladenov I., Hadzhilazova M., Djondjorov P. and Vassilev V.,
On the Plane Curves whose Curvature Depends on the Distance
from the Origin, AIP Conf. Proc. vol. 1307, American
Institute of Physics, New York 2010, pp. 112-118.

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