闪电代写 -代写CS作业_CS代写_Finance代写_Economic代写_Statistics代写_代码代做_IT代写_加急帮助

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Foundations of Applied Mathematics

MATH 4700 – Fall 2022

Homework 2

1. (30 points plus 10 bonus points) Use perturbation theory to ﬁnd two diﬀerent approximate solutions of order unity to each of the following equations:

(a) 北2 + (2 + e)北 _ 8 = 0

(b) 北2 + e2 ^2 + 北 = 3 cos e北

where e is a small parameter. Your ﬁnal solution for each root should be in the form of a main term of order unity plus one nonzero correction term, along with the usual formal estimate for the magnitude of the error of the approximation.

For equation (1a), compute the exact solution, and verify the approximation you ob- tained from perturbation theory by comparing it against the exact solution. For 10 bonus points, compare the approximation from perturbation theory against numeri- cal solution of Eq. (1b) and comment on your observations.

2. (70 points plus 25 bonus points) Consider a spacecraft which is in orbit about a planet. The equations describing its motion are Newton’s laws, which in radial

coordinates read:

_ r z 、2 = _ ,

d2 θ dr dθ

dt2 dt dt

Figure 1: Schematic of spacecraft in orbit about a planet, viewed from above the north pole of the planet.

The variables and parameters appearing here have the following meaning:

● t is time

● r = r(t) denotes the distance of the spacecraft from the center of the planet.

● θ = θ(t) denotes the angular position of the spacecraft; it ranges from 0 to 2π radians as the spacecraft traces a complete orbit. It suﬃces for simplicity to consider the spacecraft as always ﬂying above the equator of the planet; the angle θ then just denotes its angular position from the point of view of an observer looking down from the North Pole.

● M is the mass of the planet

● G is the universal gravitational constant (not the same as the gravitational con- stant g used for gravitational eﬀects near the surface of the earth)

I’ve extended you the courtesy of canceling out the factors of the mass of the spaceship from the problem so you don’t have to fuss with this irrelevant parameter.

(a) (5 points) Suppose the spacecraft is in an exact circular orbit moving counter- clockwise from the point of view of an observer looking down from the North Pole, always at the same distance R from the center of the planet. Compute how r(t) and θ(t) behave for this type of orbit.

(b) (20 points) Now suppose that a spacecraft has been ﬂying for a while in this exact circular orbit and at time t1 the spacecraft ﬁres its rockets for a short time in the forward direction. This has the eﬀect of slowing down its rotational velocity r (suddenly) by an amount u but not changing its radial velocity . Moreover, assume the rockets are on for such a short time that the position of the rocket moves only a negligible amount during the rocket ﬁring. (For those who understand the terminology, this means the same thing as treating the eﬀect of the rockets as an instantaneous impulse). After the rockets ﬁnish ﬁring, the spacecraft’s trajectory will be somewhat diﬀerent than its original orbit; in fact it may not be circular anymore. In any case, the equations (1) describe the motion of the spacecraft after the rockets ﬁnish ﬁring. We want to use these equations to predict how the spacecraft will behave over a few orbits after the rocket ﬁring. Nondimensionalize the equations of motion (1) in an intelligent way to set up for a perturbation theory calculation of the spacecraft’s motion after the rockets have ﬁnished ﬁring. You should supply an appropriate and complete set of initial conditions to be applied at the time tf at which the rockets have ﬁnished ﬁring, and nondimensionalize these intelligently as well.

Note that for full credit, you should express the equations of motion and initial conditions in completely nondimensional terms, with no dimensional quantities remaining. The variables should be properly normalized to prepare for a successful perturbation theory. All new variables and parameters you introduce should be clearly deﬁned in terms of the original variables and parameters.

(c) (10 points) Why is your choice of nondimensional variables a good one for prepar- ing for perturbation theory? What might go wrong if you had nondimensionalized them in a diﬀerent way?

(d) (10 points) When what quantity is small can a perturbation theory be pursued meaningfully to produce an approximate solution for the spacecraft’s motion after the ﬁring of the rockets? Interpret in physical terms what it means for this quantity to be small.

(e) (25 points) Perform a perturbation theory to compute the motion of the space- craft after the ﬁring of the rockets, including a main term, one nonzero correction term arising from the rocket impulse, and an estimate of the error of your approx- imation. In particular, what is the size of the ﬂuctuations of the resulting orbit about the original circular orbit?

(f) (25 bonus points) Solve the equations for the spacecraft’s motion exactly, and show that the results from your perturbation theory are in agreement with the exact solution.