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The symplectic character of orbits in the planar Magnetic-Binary problem

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Summary

The authors' aim is to present in this article the symplectic character of the planar motion of a charged particle moving in a Magnetic-Binary system. This is done by considering the classical electrodynamical formulation of the problem instead of the Hamiltonian formulation to which as known is related the concept of the symplectic matrix on which that this work is based. Treating the properties of this matrix we derived some relations between the variations of the solutions, convenient for accuracy control of any numerical work regarded with the orbit investigation of the problem.

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Abbreviations

α i ,β i ,γ i ,i=1, 2:

The directional cosines of the magnetic momentsM i ,i=1, 2 with respect to the axes of the synodic coordinates system 0xyz

λ:

The ratio of the magnitudes of the magnetic moments, i.e. λ=M 2/M 1

μ:

The reduced mass

I i ,i=1, 2:

The moments of inertia of the primaries given as the difference of their axial and equatorial moments of inertia [2]

ω:

The mean motion of the primaries revolution around their center of mass

r=(x,y,z):

The position-vector of the charged particle in the system 0xyz originated at the center of mass

References

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  3. Mavraganis, A. G.: The symplectic property of matrizants in Störmer's problem. Astrophys. Space Sci.37, 365–369 (1975).

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  4. Mavraganis, A. G.: The magnetic-binaries problem: Motion of a charged particle in the region of a magnetic-binaries system. Astrophys. Space Sci.54, 305–313 (1978).

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  5. Mavraganis, A. G.: The stationary solutions and their stability in the magnetic-binary problem, when the primaries are oblate spheroids. Astron. Astrophys.80, 130–133 (1979).

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Authors and Affiliations

  1. National Technical University of Athens, Athens, Greece

    T. J. Kalvouridis & A. G. Mavraganis

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  1. T. J. Kalvouridis
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  2. A. G. Mavraganis
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Dots over a variable represent differentiation with respect to the timet.

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Kalvouridis, T.J., Mavraganis, A.G. The symplectic character of orbits in the planar Magnetic-Binary problem. Acta Mechanica 42, 135–141 (1982). https://doi.org/10.1007/BF01176519

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