03/01/1995 09:00 AM
Computer Science
Consider a three-body system, such as an asteroid moving along a path dictated by Jupiter and the Sun. We summarize the dynamics of this system in a single matrix which is symplectic. We have written software which automates the tedious process of calculating the normal form of a symplectic matrix. Using this software, we have computed normal forms for hundreds of symplectic matrices. Using these examples, we have compiled a table of particularly nice Hamiltonian systems, uncovered which of these systems are conjugate, corrected an error in the published algorithm, formulated a new conjecture, and computed normal forms for several three-body systems which occur in nature. The symplectic normal form algorithm (first published by Burgoyne and Cushman in 1973) requires exact arithmetic. When coded in software, the algorithm must, for example, calculate (sqrt(2) *sqrt(2) - 1) - 1 as the integer zero. Using traditional software, the above expression evaluates to something closer to 11/25,000,000,000,000,000. We overcame this hurdle by applying results from field theory. In particular, we use the relationship between Q-bar and polynomial quotient space. Although we developed this system to solve the symplectic normal form problem, it can be applied to many problems which require exact arithmetic. Notes: Ph.D. Thesis
