RUI: Questions on Finiteness and Stability in Celestial Mechanics

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Proposal: DMS - 0708741 PI: Roberts, Gareth EInstitution: College of the Holy CrossTitle: RUI: QUESTIONS ON FINITENESS AND STABILITY IN CELESTIAL MECHANICSAbstractThis project investigates some well-known finiteness and stability questions in celestial mechanics. Particular attention will be given to the question on finiteness of relative equilibria equivalence classes, linear stability of relative equilibria and Saari's conjecture, that the only solutions in the n-body problem with a constant moment of inertia are relative equilibria. These questions will be approached using modern tools from algebraic geometry such as Grobner bases and BKK theory. Analytic and numerical techniques from the theory of differential equations and dynamical systems will also be employed. The problems considered here are easily generalizable to other fields of study such as geometric mechanics, the motion of point vortices, the motion of a generalized rigid body and power-law potential systems depending only on the mutual distances between bodies.The n-body problem concerns the motion of celestial bodies interacting through gravitational attraction. One of the most important types of solutions are periodic in nature, returning to their initial configuration after some fixed amount of time. Among this class of solutions, analyzing the structure and stability of simple, rigidly rotating orbits, known as relative equilibria, leads to a greater understanding of the complexities in the full problem. The study of relative equilibria is particularly useful for plotting spacecraft trajectory and discovering inexpensive methods of exploring space. Moreover, locating stable solutions provides key information pertaining to the kinds of orbits we expect to see in the universe. The educational impact of this project includes the continued mentoring of undergraduate researchers and the creation of a capstone seminar integrating the fields of celestial mechanics and algebraic geometry.