RUI: Investigating Central Configurations in the N-Body and N-Vortex Problems

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The goal of this project is to further develop the study of central configurations, an important and active sub-field of celestial mechanics. A central configuration is a special set of positions where the force on each body due to gravity points in the opposite direction of that body's position vector (with respect to the center of mass). Finding a central configuration involves solving a complicated system of nonlinear algebraic equations. Central configurations are important since they lead to families of homographic and homothetic solutions in both the N-body and N-vortex problems. Specific aims of the project include classifying five-body co-circular central configurations, studying relative equilibria in the four and five-vortex problems, using symmetry to simplify the study of central configurations for special choices of the masses, analyzing the linear stability of the corresponding relative equilibria, and describing certain symmetric families of solutions. These topics will be explored using a combination of analysis and modern and computational algebraic geometry.The N-body problem concerns the motion of celestial bodies (stars, planets, asteroids, even spaceships) interacting under their mutual gravitational attraction. Although inherently a mathematical discipline, applications to astronomy and spacecraft transport are plentiful. For example, the recent astronomical discovery of an asteroid at a ``Trojan point'' in the Earth-Sun system came centuries after the mathematical work of Lagrange on three-body central configurations. The N-vortex problem is a widely used model for understanding vorticity evolution in fluid dynamics. Some of the most important types of solutions in these problems are periodic in nature, returning to their initial configuration after some fixed amount of time. Among this class of solutions are simple, rigidly rotating orbits, known as relative equilibria, where the size and shape of the configuration is unchanged throughout the motion. Locating a relative equilibrium requires first finding a central configuration. Analyzing the structure and stability of relative equilibria leads to a greater understanding of the complexities in the full problem. In celestial mechanics this study is useful for plotting spacecraft trajectories and for discovering inexpensive methods of exploring space. In addition, locating stable solutions provides key information about the orbits astronomers and other researchers expect to see in the universe. An important priority of the project is to mentor, support and collaborate with undergraduate researchers interested in the field.