Chaos & N-Body Problems Newlin Weatherford

Newton’s Sun-Earth-Moon System Addresses problem basics in Principia (1687) Assumes: Elliptical lunar orbit Perturbations due to sun Shows sun causes the observed precession of Moon’s apogee 1694: Returns to problem, ultimately cannot solve Newton published his revised theory on lunar position in 1702, and reached a predictive margin of error of ~10 arcminutes.

Progress in 1700s 1747: Alexis Clairaut & John d’Alembert use differential equations and successive approximations Frenchman Clairaut wins St. Petersburg Academy prize for his work Predicted 1759 passing of Haley’s comet to within a month.

More Modern Developments 1887: Henri Poincaré is proves N-Body problems are chaotic. 1912: Karl Sundman finds a general*, analytic power series solution 1990: Qiudong Wang generalizes Sundman’s results* *Mostly general: cannot have singularities (direct collisions) – i.e. by starting out with 0 angular momentum

Modern Approach to N-Body Systems Sundman’s & Wang’s solutions converge very slowly — useless in practice Requires 108,000,000 terms for usage in astronomy (Beloriszky, 1930) Need numerical methods. Modern researchers - like me! - use supercomputers for large astrophysical simulations

A Mathematical Overview Note that this set of differential equations is non-linear Adapted from Goldstein, Poole, & Safko, Classical Mechanics, 3rd Ed. 2001

Adapted from Hestenes, New Foundations for Classical Mechanics, 1999, Fig. 5.1 via Goldstein, Classical Mechanics, 3rd Ed. 2001 EOMs

Example For 3 Bodies  EOMs!

Still …. Only 16 Stable Solution Sets Found for N = 3+ Here are a few:

Making A 4-Body Simulation Need more than just the EOMs for a simulation! Numerically solve the 3N = 12 simultaneous equations Set 6N = 24 initial conditions Transform gen. coords. back: animation in x,y,z My Binary Exchange Animation in Mathematica

So How Did I Actually Do It? Want meaningful gen. coords. Choose a reference frame Set an orbital inclination, true anomaly, and radial distance for each body Determine L = T – U (I chose to do this using lots of matrices and reference frame transformations)

Reference Frame Transformations Prove using matrix exponentials and Taylor expansion (just like with spin operators in QM) Rotation about x-axis Negative rotation about y-axis Rotation about z-axis

Deriving Lagrangian & Hamiltonian KE in transformations, PE: NDSolve the system of 12 ELEs w/ 24 InitCons Check that Hamiltonian is essentially constant: Hamiltonian doesn’t diverge

Slight Change in 1 of 24 InitCons Large effect from small perturbation = Chaotic Behavior

Chaotic Behavior of Trajectories due to Small Perturbation

Further Goals & Questions Add in scattering theory/statistics, maybe use impact parameters for some gen. coords. Look more into Poincaré’s original N-Body research and how it helped lead to the development of chaos theory Missing some of the “chaotic” elements? Questions?