Celestial Mechanics on a Graphing Calculator
For the restricted 2-body problem, some important simplifications can be introduced in the equations expressing Newton's laws.
We are left with the following system of differential equations, where x and y are the components of the position vector x of the smaller body, and v,w are the corresponding components of its velocity vector v. Note that d, the distance between the two bodies, is now (x^{2}+y^{2})^{1/2}.
dx -- = v dt |
dv -x -- = --------- dt (x |
dy -- = w dt |
dw -y -- = --------- dt (x |
Euler's method (Leonhard Euler, 1707-1783) is the most elementary numerical way of attacking a system of differential equations. The algorithm takes as input the equations, the length of time T over which a solution is to be approximated, the number n of steps desired for the approximation, and initial values x_{0}, y_{0}, v_{0}, w_{0} for the four unknown functions.
The method consists in iterating n times the following calculation, starting with i = 0; dt is the time increment T/n.
x |
-x |
y |
-y |
Unfortunately, while in theory Euler's method can give any desired accuracy provided the number of steps is big enough, that number of steps can be impractically large. For example, here is what happens when one applies Euler's method on a TI-82 graphing calculator to the restricted 2-body problem, as given above, with x_{0}=2, y_{0}=0, v_{0}=0, w_{0}=.68, T=20 and n=1000.
The ellipse does not close up. Doubling the number of steps takes twice as long but still does not give a closed image.