1550078481-Ordinary_Differential_Equations__Roberts_

494 Ordinary Differential Equations

Display a graph for the concentration of pollution in all lakes for both cases

and compa re the results.

In each case, how long does it take for the concentration of pollution in Lake

Ontario to b e reduced to .3%? Which la ke has its level of pollution reduced

to .25% first? last?

10.11 The Restricted Three-Body Problem

The general n-body problem of celestial mechanics is to determine the po-
sition and velo city of n homogeneous spherical bodies for all time, given their
initial positions and velocities. Although no general closed form solution to
this problem exists, much is known regarding various special cases when n is
small. The two-body problem for spheres of finite size was solved by Isaac
Newton in about 1685. A discussion of this problem as well as t h e first treat-
ment of the three-body problem appears in Book I of Newton's Principia.
The restricted three-body problem considers the properties of motion of an
infinitesimally small body when it is attracted by two finite bodies which re-
volve in circles about their center of mass and when the infinitesimally sm all
body remains in the plane of motion of the other two bodies. This problem
was first discussed by Joseph Louis Lagrange (1736-1813) in his prize winning
memoir Essai sur le Problme des Trois Corps, which he submitted to the P a ris
Academy of Sciences in 1772.

For our purposes we can think of the infinitesimally small mass as a satellite,

space-station, or spaceship and t he two large bodies as the earth and moon or

as the Sun and Jupiter. To be specific, let us consider a n earth-moon-sp aceship

system. Let E denote the mass of the earth and M denote t h e mass of the

moon. The unit of mass is chosen to b e the sum of the masses of the earth and

moon- hence, E + M = 1. It is customary to let μ represent the mass of the

small er of the two large bodies, so μ = M. The dist a nce between the two la rge

masses- E and M, in this instance-is selected as the unit oflength. The unit

of time is chosen so that the gravitational constant is l. This choice means

the two large bodies complete one circular revolution in 27!' units of time. We

will present t he equations of motion of the infinites imally small body (the

spaceship) in a special two-dimensional rectangular coordinate system- the

barycentric coordinate system. The origin of this system is at the center

of mass of the earth and moon. The x -axis passes through the gravitational

centers of the earth and moon. In t his coordinate system, the earth is located

at (-μ, 0) and the moon is located at (1-μ , 0). The location of the spaceship

is (x(t), y(t)). See Figure 10 .27.