1550078481-Ordinary_Differential_Equations__Roberts_

498 Ordinary Differential Equations

EXERCISES 10.11

1. Verify that ((1 - 2μ)/2, 0, ±v'3/2, 0) are critical points of system (2).


2. Find the y 1 -coordinate of L 1 , L2, and L3 for the Sun-Jupiter system given

that μ = .001. (The Trojan asteroids drift around the stable critical points

L4 and L s of the Sun-Jupiter system.)

3. Generate numerical solutions to system (2) on the interval [O, 10] for the
   initial conditions:

a. Near L 1 with zero velocity: y2(0) = y3(0) = y4(0) = 0

(i) Y1(0) = - 1.1 (ii) Y1(0) = - .9

b. Near L2 with zero velocity: Y2(0) = y3(0) = y 4 (0) = 0

(i) Y1 (0) = .8 (ii) Y1 (0) = .9

c. Near L3 with zero velocity: Y2(0) = y3(0) = y 4 (0) = 0

(i) Y1 (O) = 1.1 (ii) Y1 (O) = 1.2

d. Near L4 with zero velocity: Y1(0) = .5, Y2(0) = 0, y3(0) = .9, y4(0) = 0

In each case, display a phase-plane graph of y 3 versus y 1 (y versus x). What
happens to the spaceship in each instance? (All five points L 1 , L2, L3, L 4 ,
and Ls have been considered as sites for locating permanent space stations.
Which locations do you think would be better? Why?)

4. In this example a spaceship is init ia lly on the side of the earth opposite the
   moon. The pilot shuts off the rocket engines to begin a non-powered flight.
   The purpose of the exercise is to see the effects of "burnout" position and
   velocity on the trajectory of the spaceship. Numerically solve system (2) with
   μ = .012129 on the interval [O, 50] for the following three initial conditions:

a. Y1(0) = -1.2625, Y2(0) = 0, y3(0) = 0, y4(0) = 1.05

b. Y1(0) = -1.26, Y2(0) = 0, y3(0) = 0, y4(0) = 1.05

c. Y1(0) = -1.2625, Y2(0) = 0, y3(0) = 0, y4(0) = 1.00

For the initial conditions a., b., and c. display a phase-plane graph of y 3 versus
Y1. Compare the three graphs. What do you conclude? Is the initial position
of the spaceship very important? Is the initial velocity of the spaceship very
important?

(NOTE: The specified accuracy of the numerica l integration technique is
very important a lso. Use a good double precision integration routine to solve
system (2) with a prescribed accuracy of at least 10-^12 .)