ADVANCED DYNAMICS — PHY 4241/5227

HOME AND CLASS WORK – SET 12

(March 12, 2010)

(42) Derive the Taylor expansions of the functions exp(x), cosh(x), cos(x), sinh(x) and sin(x) about x = 0 from the equation

f(x) = ∑_{n=0}^{∞} f^{(n)}(0) \frac{x^{n}}{n!}

by calculating f^{(n)}(0) for each case explicitly. Do NOT use Euler’s formula. Due April 2 before class (5 points).

(43) The matrix L is defined by

L = ( \begin{matrix}l_{0}^{0} & l_{1}^{0} & l_{2}^{0} & l_{3}^{0} \\ l_{0}^{1} & l_{1}^{1} & l_{2}^{1} & l_{3}^{1} \\ l_{0}^{2} & l_{1}^{2} & l_{2}^{2} & l_{3}^{2} \\ l_{0}^{3} & l_{1}^{3} & l_{2}^{3} & l_{3}^{3} \end{matrix} )

1. Calculate −g L (2 points).

2. Write down the transpose matrix L̃ (2 points).

3. Calculate L̃ g (2 points).

4. Compare 1. and 2. to find the general form of L (i.e. use L̃ g = −g L, 4 points).

5. Obtain the same result by discussing the elements of the equation g^{αβ}l̃_{β}^{γ}g_{γδ} = −l_{δ}^{α} (4 extra points. Hint: Do the contractions first.)

Due April 7 in class.

(44) In the following K_{1} and S_{3} are generators as defined in class.

1. Calculate exp(−ζ K_{1}) and explain its physical meaning (3 points).

2. Calculate exp(−ϕ S_{3}) and explain its physical meaning (3 points).

Due April 12 before class.

(45) Consider the spaceship journey again and plot versus the proper time τ the following quantities (due April 5 before class, 10 points):

1. The time on earth at which the spaceship, seen from earth, is at its position at time τ.

2. The distance from earth as seen from earth.

3. The date of news received from earth, which is transmitted at the speed of light.