Numerical Integration, Lyapunov Exponents, and the Outer Solar System
Monday, October 29, 2001 - 11:00am - 12:00pm
The numerical integration of ordinary differential equations introduces errors that can fundamentally alter the nature of the computed solution, a point eloquently made by Mitchell Feigenbaum in his development of the quadratic map. Hamiltonian systems offer some extraordinary challenges to our ability to quantify the uncertainty present in our representation of such systems. In this lecture, I will review some of the fundamental issues germane to the numerical integration of ordinary differential equations, including Hamiltonian systems. In particular, we will focus on the properties of so-called symplectic methods, showing their significance to Hamiltonian problems, as well as their limitations. In particular, we will explore how computed trajectories will differ from their exactly calculated counterparts as a function of step size, and focus on how symplectic systems that utilize inappropriately chosen step sizes can manifest artificial chaotic behavior. Our presentation will focus on analytic examples drawn from classical mechanics---including the harmonic oscillator and the pendulum---as well as computational examples drawn from celestial mechanics---including the behavior of the outer solar system.