User:Egm6322.s14.team2/report4

=Report 4 PEA 2 Spring 2014 Team 2=

Problems are given in lecture notes

Problem Statement
(a): Show

(b): Obtain directly Equation 1-2 from Equation 1-1 and one other equation.

(c): Prove

(d): Plot $${\alpha}=sin{\phi}$$ for varios values of e (e.g. e=1/4, 1/2, 3/4) and ω (ω=1/2 and 3/2).

Solution

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Part a
Using polar coordinates, the unit vectors that form a basis in the orbital plane and their first order time derivatives are:

The position vector, r, in this basis is:

The time derivative of the position vector is the velocity:

The second derivative with respect to time is the acceleration:

After combining like terms and factoring a $$\frac{1}{r}$$ out of the $$e_{\theta}$$ term, the equation above simplifies to:

The second term in the expression for the acceleration vector in ($$) can be re-written in a different form by utilizing the definition of the product rule from calculus.

Therefore with the substitution from ($$) made into ($$) the following result is obtained.

Part b
With all the force in the normal direction and F equal to the change in momentum ($$F=\frac{d}{dt}(m{\mathbf{v}})$$), the tangential component of the acceleration, the change in angular momentum, is zero.

Part c
Using the equations for an ellipse and a circle, the values of y as a function of x are:

The area of the corresponding segment and triangle:

After rearranging, Equation 20 becomes:

Part d
In this section the plots of $$ \alpha $$ are presented for various values of $$ \omega \text{ and } e $$.











Noticed how in the plots, as the value of e increases, the peaks of the plots move to the left.

Problem Statement
This problem is motivated by notes provided by Dr. Vu-Quoc. The problem refers to Pb.70.[5,6,7] in Sec. 70b.

(a): Show that Equation 2-1 is an odd, periodic function of μ with period 2π.

(b): Show

(c): Plot φ(t) and μ(t) for τ=0, e-1/2, ω=1/2, and t [0,5T]. Find the true anomaly, $$ \theta (t) $$, and plot it on the same plot with $$\phi (t)$$ and  $$\mu (t)$$.

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Part a
Taking the first derivative of ($$) with respect to $$ \phi$$

Since $$\sin{\phi}$$ is bounded between -1 and 1, ($$) is positive definite when $$|e| \le 1$$.

Since ($$) is positive definite, it is a monotonically increasing function of $$\phi$$, and is one to one, invertible function.

A function is odd if:

It is clearly seen that $$ f(\phi) $$ is an odd function of $$\phi$$

There is an easily proved theorem that states that a function is odd, then it's inverse is also odd. Function_is_Odd_Iff_Inverse_is_Odd.

Thus, $$\phi(\mu)=f^{-1}(\mu) $$ is an odd function of $$\mu$$

Adding or subtracting $$2 \pi $$ to the argument in ($$)

Since $$\sin{\mu}$$ is periodic with period $$2\pi$$:

Plugging ($$) into ($$):

Applying $$f^{-1}$$ to each side:

Using the definition in ($$)

It can now be shown that $$\phi-\mu $$ is a periodic function of $$\mu$$

Plugging ($$) into ($$) and rearranging:

Adding or subtracting $$2 \pi $$ to the arguments in ($$)

Using ($$),

and using fact that $$ \sin{(\phi)} $$ is periodic with period of $$ 2 \pi $$, ($$) reduces to:

Therefore:

Next, it can be shown that $$\phi-\mu $$ is an odd function of $$\mu$$

Negating the argument in ($$):

It has already been shown that $$ f^{-1}(\mu)$$ is odd and that the sine function is odd; using the property of odd functions:

And finally:

Thus, $$\phi(\mu)-\mu$$ is an odd, perioid function with period $$2 \pi$$.

Part b
In this section we are asked to show that ($$) is correct.

We start by showing what we are working with.

From the first term on the LHS of ($$) we can split the integral up.

The integral of a cosine function in the second term on the RHS of ($$) is zero. Therefore the integral on the LHS of ($$) can be expressed in the following simplified form.

The differentials $$ d{\mu} $$ cancel out from the RHS expression in ($$) leaving the following.

The variable $$ {\mu} $$ can be expressed in terms of the variable $$ {\phi} $$ from ($$. Making this substitution yields.

We must express $$ \mu $$ in terms of $$ \phi $$ and change the bounds of integration to complete the expression. To do this we turn to ($$) to substitute for $$ \mu $$. We also utilize ($$) to find out what the value of $$ \phi $$ is when $$ \mu $$ takes on specific values of $$ 0 \text{ and } \pi $$. Doing so yields the following.

At this point we can express ($$) in the following form.

Where:

Part c
Here we are asked to plot $$ \phi \text{ and } \mu $$ as functions of time. The results are presented below. Please note that the function $$ \mu $$ is linear with respect to time, so it is the line on the plots below.

The following relation was used to compute the values of the mean, eccentric, and true anomaly.

The analytic relation given by ($$) was found on a Wikipedia page about the True Anomaly. I do not know how that expression was derived.



Some observations about the figure above. As the angular frequency is increased, the lines tilt closer to the vertical axis. Notice how the plots for $$ \theta (t) $$ stay long the bottom of the plots. This is most likely due to the fact that an inverse cosine relation had to be used to compute the value, and as such the output of such a function is limited between 0 and $$ \pi$$.



Problem Statement
(a): Verify that Equations 3-1 through 3-5 are homogeneous solutions of the Legendre equation.

(b):Show that Equation 3-6 can also be written as Equation 3-7.

(c): Verify that Equations 3-1 through 3-5 can be obtained from Equation 3-6 or 3-7.

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Part a
The Legendre differential equation is

The first and second derivatives are:

P0=1 is the solution to the zero order Legendre differential equation (i.e. n=0):

Substituting P0 and its first and second derivatives in the the Legendre differential equation:

For n=1:

The first and second derivatives are:

Substituting P1 and its first and second derivatives in the the Legendre differential equation:

For n=2:

The first and second derivatives are:

Substituting P2 and its first and second derivatives in the the Legendre differential equation:

For n=3:

The first and second derivatives are:

Substituting P3 and its first and second derivatives in the the Legendre differential equation:

For n=4:

The first and second derivatives are:

Substituting P4 and its first and second derivatives in the the Legendre differential equation:

Part b
By examination, the numerator in Equation 3-6 includes the product of the odd terms of the expression (2n-2i)! in the numerator of Equation 3-7. The (2n-2i)/2 even terms can be expressed as follows:

Substituting the difference of (2n-2i)! and the even terms in Equation 3.29 for the terms for the odds in Equation 3-7, yields:

Adding the exponents of 2 and rearranging, gives Equation 3-6.

Part c
Solving Equation 3-31 for n=1, 2, 3, and 4, gives Equations 3-1 through 3-5.

Problem Statement
(a): Verify

{{NumBlk|::|$$ \int_{0}^{T}e^{i(n-m){\omega}{\theta}}d{\theta}= \left\{\begin{matrix} 0 & \text{if } n\neq m\\ T& \text{if }n=m \end{matrix}\right. $$ |$$|}}

Where,

(b): Relate the complex Fourier coefficient:

To the Fourier coefficients given by:

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Part a
We want to compute the integral in ($$):

Let us first examine the case where $$n \ne m$$:

Multiplying by the complex conjugate:

Using Euler's Identity:

($$) then becomes:

For the case where $$ n=m, n-m=0 $$, ($$) then becomes:

Part b
There are three cases to examine:

Case 1: $$ n > 0; -i n \omega \theta < 0$$ After applying Euler's identity, ($$) becomes:

Using $$) and $$),

Case 2: $$ n < 0; -i n \omega \theta > 0$$ After applying Euler's identity, ($$) becomes:

Using $$) and $$),

Case 3: $$ n = 0; -i n \omega \theta = 0$$

($$) becomes:

Using ($$)

Problem Statement
R*9.3

(a): show

(b): show

(c): Find the expression of $$Q_2(x)$$ in terms of $$\tanh{^{-1}}(x)$$

R*9.4

(d): Verify the expressions of $$Q_0$$, $$Q_1$$, $$Q_2$$ using the following:

R*9.5

(e): Show that $$Q_n$$ is odd or even depending on the index $$n$$

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Part a
The hyperbolic tangent function is defined as:

Thus,

Part b
Using ($$) in ($$):

Part c
Using ($$) in ($$):

Part d
n=0:

Using ($$)

($$) becomes:

n=1: Using ($$)

($$) becomes:

n=2: Using ($$)

($$) becomes:

Part e
n=0:

Thus, $$Q_0(x)$$ is an odd function

n=1:

It has already been shown that $$\log{(\frac{1+x}{1-x})}$$ is an odd function. $$x$$ is also a function, and the product of two odd functions is an even function.

For an even function: $$f(-x)=f(x)$$

Thus, $$Q_1(x)$$ is even.

Problem Statement
Find the expression for the gradient of a scalar function in the following coordinate systems

(a):Cartesian

(b): Cylindrical

(c): Spherical

(d): Elliptical

(e): Parabolic

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Part(a): The gradient in Cartesian coordinates
The definition of the gradient in curvilinear coordinates is:

Here we are dealing with the familiar Cartesian coordinate system. It is described using orthonormal vectors.

Part (b): Cylindrical Coordinates
The coordinates take the form:

The derivatives of u1, u2, and u3 are:

The sum of the squared derivatives:

Using the Pythagorean Trigonometric Identity, the equation reduces to:

The magnitude of the tangent vector h are then identified as follows:

Substituting the tangent vectors back into the general equation for the gradient, the gradient in cylindrical coordinates is:

Part(c): Spherical Coordinates
The coordinates take the form:

The derivatives of u1, u2, and u3 are:

The sum of the squared derivatives:

Using the Pythagorean Trigonometric Identity, the equation reduces to:

The magnitude of the tangent vector h are then identified as follows:

Substituting the tangent vectors back into the general equation for the gradient, the gradient in spherical coordinates is:

Part(d): Elliptical Coordinates
The derivatives of u1 and u2 are:

Using both trigonometric identities $$sin^{2}u+cos^{2}u=1$$ and $$cosh^{2}u-sinh^{2}u=1$$:

The magnitude of the tangent vector h are then identified as follows:

Part(e): Parabolic Coordinates
The derivatives of u1 and u2 are:

The magnitude of the tangent vector h are then identified as follows:

Problem Statement
Address comments in Report 3.

Problem Statement
sec.67b, Pb-67.2 (equilibrium of line defects, Hermite diff. eq., Hermite polynomials)

Equation ($$) is given:

Find a suitable change of variable to show that ($$) ((6) p.68-5) can be written as the Hermite differential equation of the form:

For the 2 cases of $$ N = 7,8 $$ particles, find the non-dimensional equilibrium positions and plot them.

Find
(a): A suitable change of variable to show that ($$) can be written as the Hermite differential equation of the form of ($$)

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Part (a): Find Suitable Change of Variable
For this problem we seek to determine a suitable change of variables that will give us ($$) from ($$). The variable of interest is changing from u to x in this problem. We need to find some sort of relation between the two variables that will gives us what we want. We can observe the differences between the two forms of the differential equations and conclude that a factor of 2 on the zeroth and first derivative terms is an observable difference. We can also observe that the transformed differential equation is of the same order as the original, therefore the change of variables does not include a derivative term. It must therefore only contain the independent variables u and x.

Using the transformation $${\rho}(u)={\lambda}u$$ and the first and second derivatives below in the function P:

Make the substitution $$ u{\lambda}=\rho (u) $$ and then divide the equation by $${\lambda}^{2}$$ to get.

Examining equation ($$) in comparison to the the Hermite differential equation $$, $${\lambda}=\frac{1}{\sqrt{2}}$$ and $${\rho}(u)=\frac{u}{\sqrt{2}}$$.

Using y instead of P(ρ(u)) and x instead of $$ \rho (u)$$, the equation takes the form of the Hermite differential equation above.

Plot Solution for 2 Cases
Here we are asked to plot the solution to the Hermite differential equation for the cases when $$ N = 7,8 $$.

Here the differential equation takes the following form when $$ N=7 $$.

The equation for the coefficients of the Hermite polynomial can be obtained by using a power series solution to the even or odd Hermite differential equation:

Therefore, the 7th and 8th order Hermite polynomials, respectively, are:



The roots from Wolfram Alpha using the functions HermiteH[7,x] and HermiteH[8,x] are:

This equation is a second order linear differential equation that is homogenous with variable coefficients. A solution is sought in the form of a finite power series. It is finite in this case because the original polynomial that we are solving for is a finite polynomial i.e. it only contains terms up to $$ u^{N} $$

Wolfram Alpha gives the solution

R3.5: Testing Gauss Laguerre quadrature
As stated in R3.5: Dr. Burkardt code did not test a Bessel integral of the form given in problem R2.7, he does cite the reference:

Philip Davis, Philip Rabinowitz, Methods of Numerical Integration, Second Edition, Dover, 2007, ISBN: 0486453391, LC: QA299.3.D28. In his source code, but does not give a page or chapter. To date I have not been able to locate any helpful information. I will try emailing him next.

=Contributing Team Members=


 * Cameron Stewart Solved problems 2a, 4,5 and the portion of problem 7 related to R3.5. Reviewed problem 6.
 * Elizabeth Bartlett Solved problems 1, 3, 6 and the portion of problem 7 related to R3.2. Reviewed problems 4-2a and 4-5.
 * Kevin Frost Solved problem 5
 * Christopher Neal Solved problems 1d, 2b, 2c