ps11 - In particular show that the commutator algebra is...

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PHY 504 Problem Set #11 due 22 November 2010 1. Consider a system of N coupled oscillators like those in Figure 13.1, except that these oscillate transversely in the vertical direction only. Assume they are connected by springs of force constant k and that they are arranged in a horizontal circle. (a) Find the normal modes and frequencies of small oscillations. (b) Take the continuum limit to obtain a continuum Lagrangian of the form (13.6) and continuum equations of motion analogous to (13.7). (c) What is the speed of sound? Is it the same in this case for longitudinal as for transverse waves? 2. For the Lagrangian find the conserved quantity associated with simultaneous boosts of x 1 and x 2 . Explain why this conservation law amounts to conservation of center-of-mass velocity. 3. Find the commutators of the generating matrices of Lorentz transformations with each other and with the generators of rotations.
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Unformatted text preview: In particular, show that the commutator algebra is closed. 4. Problem 7.18. 5. Starting from rest, a spaceship leaves Earth and flies to Proxima Centauri, 4.2 light-years away. For the first half of the journey, the ship maintains a steady acceleration of 9.8 m/s 2 in its own reference frame. For the second half, it decelerates at the same rate. (a) What is the maximum velocity relative to Earth attained by the ship? (b) How long in ship time does it take the ship to get there? (c) How long in Earth time? 6. Problem 7.23. [Review Problem 2, PS #1 first.] What minimum fraction of its rest mass must the spaceship of problem #2 expend in order to complete its journey, assuming the rocket gases are expelled with perfect efficiency at the speed of light? Compare with the nonrelativistic (Newtonian) answer....
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