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1. Alhowaity, Sawsan Salem. Relative equilibria in the curved N-body problem.

Degree: Department of Mathematics and Statistics, 2018, Canadian Mathematical Bulletin

URL: https://dspace.library.uvic.ca//handle/1828/10037

We consider the curved N-body problem, N > 2, on a surface of constant Gaussian curvature κ ≠ 0; i.e., on spheres S2κ, for κ > 0, and on hyperbolic manifolds H2κ, for κ < 0. Our goal is to define and study relative equilibria, which are orbits whose mutual distances remain constant during the motion. We find new relative equilibria in the curved N-body problem for N = 4, and see whether bifurcations occur when passing through κ = 0. After obtaining a criterion for the existence of quadrilateral configurations on the equator of the sphere, we study two restricted 4-body problems: One in which two bodies are massless , and the second in which only one body is massless. In the former we prove the evidence for square-like relative equilibria, whereas in the latter we discuss the existence of kite-shaped relative equilibria.
We will further study the 5-body problem on surfaces of constant curvature. Four of the masses arranged at the vertices of a square, and the fifth mass at the north pole of S2κ, when the curvature is positive, it is shown that relative equilibria exists when the four masses at the vertices of the square are either equal or two of them are infinitesimal, such that they do not affect the motion of the remaining three masses. In the hyperbolic case H2κ, κ < 0, there exist two values for the angular velocity which produce negative elliptic relative equilibria when the masses at the vertices of the square are equal. We also show that the square pyramidal relative equilibria with non-equal masses do not exist in H2κ.
Based on the work of Florin Diacu on the existence of relative equilibria for 3-body problem on the equator of S2κ, we investigate the motion of more than three bodies. Furthermore, we study the motion of the negative curved 2-and 3-centre problems on the Poincaré upper semi-plane model. Using this model, we prove that the 2-centre problem is integrable, and we study the dynamics around the equilibrium point. Further, we analyze the singularities of the 3- centre problem due to the collision; i.e., the configurations for which at least two bodies have identical coordinates.
*Advisors/Committee Members: Ibrahim, Slim (supervisor), Chavela, Ernesto (supervisor).*

Subjects/Keywords: Relative Equilibria; Curved N-Body Problem; relative equilibria

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APA (6^{th} Edition):

Alhowaity, S. S. (2018). Relative equilibria in the curved N-body problem. (Thesis). Canadian Mathematical Bulletin. Retrieved from https://dspace.library.uvic.ca//handle/1828/10037

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Alhowaity, Sawsan Salem. “Relative equilibria in the curved N-body problem.” 2018. Thesis, Canadian Mathematical Bulletin. Accessed December 18, 2018. https://dspace.library.uvic.ca//handle/1828/10037.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Alhowaity, Sawsan Salem. “Relative equilibria in the curved N-body problem.” 2018. Web. 18 Dec 2018.

Vancouver:

Alhowaity SS. Relative equilibria in the curved N-body problem. [Internet] [Thesis]. Canadian Mathematical Bulletin; 2018. [cited 2018 Dec 18]. Available from: https://dspace.library.uvic.ca//handle/1828/10037.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Alhowaity SS. Relative equilibria in the curved N-body problem. [Thesis]. Canadian Mathematical Bulletin; 2018. Available from: https://dspace.library.uvic.ca//handle/1828/10037

Not specified: Masters Thesis or Doctoral Dissertation