lowest energy configurations

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    Willemijn Goossens
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    – alpha =0
    In this case, the quadrupole contribution to the gravitational energy is always zero. Since the monopole energy is independent of the angle of the dumbbell with respect to the z-axis, all orientations of the dumbbell are equal in energy.

    – alpha >0
    Rewriting -alpha(2cos^2-sin^2) as -alpha(3cos^2-1) allows to predict the behaviour of this function in terms of theta. Take alpha=1 for simplicity. Then the quadrupole correction to the energy varies between -2 for theta=0° and 1 for theta=90°. The quadrupole correction is thus most negative for theta=0°. This means that the lowest-energy configuration is the one with the dumbbell parallel to the z-axis.

    – alpha <0
    The quadrupole correction now varies between 2 for theta=0° and -1 for theta=90° (for alpha=-1), making it the most negative for theta=90°. Thus, in this case the lowest-energy orientation for the dumbbell is within the xy-plane.

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