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The angular momentum of a relative equilibrium

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  • There are two main reasons why relative equilibria of $N$ point masses under the influence of Newton attraction are mathematically more interesting to study when space dimension is at least 4:
        1) in a higher dimensional space, a relative equilibrium is determined not only by the initial configuration but also by the choice of a hermitian structure on the space where the motion takes place (see [3]); in particu\-lar, its angular momentum depends on this choice;
        2) relative equilibria are not necessarily periodic: if the configuration is balanced but not central (see [3,2,7]), the motion is in general quasi-periodic.
        In this exploratory paper we address the following question, which touches both aspects: what are the possible frequencies of the angular momentum of a given central (or balanced) configuration and at what values of these frequencies bifurcations from periodic to quasi-periodic relative equilibria do occur? We give a full answer for relative equilibrium motions in $R^4$ and conjecture that an analogous situation holds true for higher dimensions. A refinement of Horn's problem given in [12] plays an important role.
    Mathematics Subject Classification: Primary: 70F10, 15A18.


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