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The problems of minimizing the potential energy of electrons on a sphere, or maximizing the smallest distance between the electrons, have been well-studied. E.g., see the earlier MO question "Distributing points evenly on a sphere."

My question concerns the same problems on an ellipsoid with two axis dimensions $a=b$ equal, and the third dimension $c$ much shorter, so the ellipsoid is pancake-like:

          EllipsoidSaucer
          Ellipsoid axis dimensions: $a=5, b=5, c=1$, with $c$ the vertical dimension.
It is natural to expect that with $c \ll a,b$, the electrons form an approximate honeycomb hexagonal packing on the upper and lower surface of the ellipsoid, perhaps with some edge effects near the horizontal midplane.

Has this problem been studied? Are there results established under certain conditions?

Update. Here is a nice image from the Müller/Frauendiener paper that Carlo Beenakker cites:
          ElectronsTorus
          Fig.6 (detail): $1024$ charged particles on a torus.
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Joseph O'Rourke
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  • I was going to suggest capacitor design, but this is a different situation. Nevertheless, you might have some success asking this in an electrical engineering forum, even if they are less concerned with precision. Gerhard "Or Do I Mean Accuracy" Paseman, 2016.02.27. – Gerhard Paseman Feb 28 '16 at 03:01
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    Is the ellipsoid an isolator? Would be yet another variant under which the solutions may be different. – Manfred Weis Feb 28 '16 at 14:32
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    The answer for minimizing the potential energy (I am assuming a Coulomb potential) and maximizing the minimum distance are going to be very different. The latter gives an asymptotically uniform distribution, while the former will give an uneven distribution with much lower density on the top and bottom. The transition between asymptotic uniformity and nonuniformity is discussed in Hardin and Saff, 2005 http://dx.doi.org/10.1016/j.aim.2004.05.006 – Yoav Kallus Feb 28 '16 at 14:42
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    software that will calculate this for you is provided in Charged particles constrained to a curved surface – Carlo Beenakker Feb 28 '16 at 14:42
  • For an experimental answer, you might try to obtain such an oblate ellipsoid made metal and then apply some ferrofluid to it https://www.youtube.com/watch?v=5APHa7vscoI – Manfred Weis Feb 28 '16 at 14:44
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    A little Googling came up with this homework problem(?): (http://www.physics.princeton.edu/~mcdonald/examples/ellipsoid.pdf), but I haven't checked the calculation. The electrostatic calculation gives the asymptotic density when the number of electrons is very large. – Yoav Kallus Feb 28 '16 at 14:54
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    The continuous case has been solved for a more general class of shapes that includes oblate ellipsoids. See I W McAllister 1990 J. Phys. D: Appl. Phys. 23 359 doi:10.1088/0022-3727/23/3/016 and http://math.stackexchange.com/questions/112662/gaussian-curvature-of-an-ellipsoid-proportional-to-fourth-power-of-the-distance . –  Feb 28 '16 at 15:21
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    The perfectly flat pancake (a circular disk) could also be considered, as the limiting case. What would the configurations of a reasonably small numbers $k$ of electrons look like? One can predict some configurations for $k\le 7$ maybe a few beyond $7$. At which value of $k$ lack of symmetry occurs first? – Wlodek Kuperberg Jan 02 '17 at 00:06

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