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Central Configurations of the Curved N-Body Problem

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Abstract

We consider the N-body problem of celestial mechanics in spaces of nonzero constant curvature. Using the concept of effective potential, we define the moment of inertia for systems moving on spheres and hyperbolic spheres and show that we can recover the classical definition in the Euclidean case. After proving some criteria for the existence of relative equilibria, we find a natural way to define the concept of central configuration in curved spaces using the moment of inertia and show that our definition is formally similar to the one that governs the classical problem. We prove that, for any given point masses on spheres and hyperbolic spheres, central configurations always exist. We end with results concerning the number of central configurations that lie on the same geodesic, thus extending the celebrated theorem of Moulton to hyperbolic spheres and pointing out that it has no straightforward generalization to spheres, where the count gets complicated even for two bodies.

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Acknowledgements

Cristina Stoica and Florin Diacu enjoyed partial support from Discovery Grants awarded by NSERC of Canada. Shuqiang Zhu was funded by a University of Victoria Scholarship and a David and Geoffrey Fox Graduate Fellowship.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Victoria, Victoria, Canada

    Florin Diacu & Shuqiang Zhu

  2. Yale-NUS College, National University of Singapore, Singapore, Singapore

    Florin Diacu

  3. Department of Mathematics, Wilfrid Laurier University, Waterloo, Canada

    Cristina Stoica

  4. School of Mathematical Sciences, University of Science and Technology of China, Hefei, China

    Shuqiang Zhu

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  1. Florin Diacu
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  2. Cristina Stoica
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  3. Shuqiang Zhu
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Correspondence to Shuqiang Zhu.

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Communicated by Tudor Stefan Ratiu.

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Diacu, F., Stoica, C. & Zhu, S. Central Configurations of the Curved N-Body Problem. J Nonlinear Sci 28, 1999–2046 (2018). https://doi.org/10.1007/s00332-018-9473-y

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