1. Chaos & N-Body Problems
Newlin Weatherford

2. 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.

3. 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.

4. 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

5. 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

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

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

8. Example For 3 Bodies 
EOMs!

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

10. 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

11. 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)

12. 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

13. 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

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

15. Chaotic Behavior of Trajectories due to Small Perturbation

16. 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?