Central configurations of the curved N-body problem

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dc.contributor.author Zhu, Shuqiang
dc.date.accessioned 2017-07-14T19:59:30Z
dc.date.available 2017-07-14T19:59:30Z
dc.date.copyright 2017 en_US
dc.date.issued 2017-07-14
dc.identifier.uri https://dspace.library.uvic.ca//handle/1828/8334
dc.description.abstract We extend the concept of central configurations to the N-body problem in spaces of nonzero constant curvature. Based on the work of Florin Diacu on relative equilib- ria of the curved N-body problem and the work of Smale on general relative equilibria, we find a natural way to define the concept of central configurations with the effective potentials. We characterize the ordinary central configurations as constrained critical points of the cotangent potential, which helps us to establish the existence of ordi- nary central configurations for any given masses. After these fundamental results, we study central configurations on H2, ordinary central configurations in S3, and special central configurations in S3 in three separate chapters. For central configurations on H2, we generalize the theorem of Moulton on geodesic central configurations, the theorem of Shub on the compactness of central configurations, the theorem of Conley on the index of geodesic central configurations, and the theorem of Palmore on the lower bound for the number of central configurations. We show that all three-body central configurations that form equilateral triangles must have three equal masses. For ordinary central configurations in S3, we construct a class of S3 ordinary central configurations. We study the geodesic central configurations of two and three bodies. Three-body non-geodesic ordinary central configurations that form equilateral trian- gles must have three equal masses. We also put into the evidence some other classes of central configurations. For special central configurations, we show that for any N ≥ 3, there are masses that admit at least one special central configuration. We then consider the Dziobek special central configurations and obtain the central con- figuration equation in terms of mutual distances and volumes formed by the position vectors. We end the thesis with results concerning the stability of relative equilibria associated with 3-body special central configurations. We find that these relative equilibria are Lyapunov stable when confined to S1, and that they are linearly stable on S2 if and only if the angular momentum is bigger than a certain value determined by the configuration. en_US
dc.language English eng
dc.language.iso en en_US
dc.rights Available to the World Wide Web en_US
dc.subject Hamiltonian system en_US
dc.subject celestial mechanics en_US
dc.subject geometric mechanics en_US
dc.subject curved N-body problem en_US
dc.subject central configurations en_US
dc.subject Morse theory en_US
dc.subject stability en_US
dc.title Central configurations of the curved N-body problem en_US
dc.type Thesis en_US
dc.contributor.supervisor Diacu, Florin
dc.degree.department Department of Mathematics and Statistics en_US
dc.degree.level Doctor of Philosophy Ph.D. en_US
dc.identifier.bibliographicCitation S. Zhu, Eulerian Relative Equilibria of the Curved 3-Body Problems in S2, Proc. Amer. Math. Soc. 142 (2014), no. 8, 2837-2848. en_US
dc.identifier.bibliographicCitation S. Zhu, S. Zhao, Three-Dimensional Central Configurations in H3 and S3, J. Math. Phys. 58 (2017), no. 2, 022901. en_US
dc.identifier.bibliographicCitation F. Diacu, J.M. Sa ́nchez-Cerritos, S. Zhu, Stability of Fixed Points and Associated Relative Equilibria of the 3-Body Problem on S1 and S2, J. Dynam. Differential Equations (2016). doi:10.1007/s10884-016-9550-6. en_US
dc.identifier.bibliographicCitation E. Boulter, F. Diacu, S. Zhu, The N-Body Problem in Spaces with Uni- formly Varying Curvature, J. Math. Phys. 58(2017), no. 5, 052703. en_US
dc.identifier.bibliographicCitation F. Diacu, C. Stoica, S. Zhu, Central Configurations of the Curved N-Body Prob- lem, to appear in J. Nonlinear Sci., arXiv:1603.03342. en_US
dc.description.scholarlevel Graduate en_US

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