1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations

(x(t), y(t))

-μ 1-μ

Figure 10.27 The Restricted Three-Body Problem

and Associated Critical Points

495

x

Using Newton's second law of motion and the inverse square law of motion,

it can be shown that the equations of motion for the spaceship are

x" = 2 y' + x (1 - μ)(x + μ) μ(x - 1 + μ)
r3 s3
(1)
y" = -2x' + y - (l - μ)y - μy
r3 s3
where r = ((x + μ)^2 + y^2 )^112 and s = ((x - 1 + μ)^2 + y^2 )^112. Thus, r is
the distance of the spaceship from the earth and s is its distance from the
moon. If we let Y1 = x, Y2 = x', y3 = y, and y4 = y', we obtain the following
equivalent system of four first-order equations

(2.1) y~ = Y2

I (1 - μ)(Y1 + μ)

(2.2) Y2 = 2y4 + Y1 - ((yi + μ)2 + y~)3/2

μ(y1 - 1 + μ)

((y1 - 1 + μ)2 + y~)3/2

(2.3) Y3 I = Y4