Central Configurations of the Curved N-Body Problem

Central Configurations of the Curved N-Body Problem J Nonlinear Sci https://doi.org/10.1007/s00332-018-9473-y Central Configurations of the Curved N-Body Problem 1,2 3 Florin Diacu · Cristina Stoica · 1,4 Shuqiang Zhu Received: 4 February 2017 / Accepted: 25 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 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. Communicated http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Nonlinear Science Springer Journals

Central Configurations of the Curved N-Body Problem

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Publisher
Springer US
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Analysis; Theoretical, Mathematical and Computational Physics; Classical Mechanics; Mathematical and Computational Engineering; Economic Theory/Quantitative Economics/Mathematical Methods
ISSN
0938-8974
eISSN
1432-1467
D.O.I.
10.1007/s00332-018-9473-y
Publisher site
See Article on Publisher Site

Abstract

J Nonlinear Sci https://doi.org/10.1007/s00332-018-9473-y Central Configurations of the Curved N-Body Problem 1,2 3 Florin Diacu · Cristina Stoica · 1,4 Shuqiang Zhu Received: 4 February 2017 / Accepted: 25 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 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. Communicated

Journal

Journal of Nonlinear ScienceSpringer Journals

Published: Jun 5, 2018

References

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