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Idea #1 for a written problem (POW):
Analyze the modes and frequencies of the oscillators of problems 3 and 4 (see below), along
with that of a 4 mass linear oscillator system (like prob 3, but with 4 masses),
and any other oscillator system you find particularly interesting.
Discuss their nature, and the relationship of their modes/dynamics to the
symmetry/nature of their underlying "Hamiltonian" (H=T+U).
Include illustrations in terms of cartoon pictures of relevant modes...

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Idea #1a for a written problem (POW):
Analyze the modes and frequencies of the oscillator of problem 4, along
with that of a 3 mass linear oscillator system where the three masses are 
constrained to move on a circular ring and connected by indentical spings.
possible extensions:
Consider the nature of larger (more masses) ring systems.
Discuss combining modes to get different types of motion and especially
motion which recovers the original symmetry of the system.
Include illustrations in terms of cartoon pictures.


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Idea # 2 for written problem:
   present a comparitive analysis of a lightly damped oscillator driven
at T=pi/w_0, 2pi/w_0 , 4pi/w_0, ... by a square wave pulse (as in the first POW).

For large T, compare the result you get from a Green's function approach to that
from a Fourier series approach.  Discuss the relationship of the two approaches.


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For reference:


2) Calculate the normal mode frequencies, eigenvectors (draw them),
for the asymmetric double mass/spring spring system (2 springs) 
<pre>

          /|  k    _   k    _
          /|zzzzzz|m|zzzzzz|m|               Asymmetric
          /|       -        -

Illustrate each mode with a cartoon or picture.
b) Write down the equation for the position of the left-hand mass as a function 
of time if it is struck at t=0 by an identical mass moving to the right at v_0.

3) Calculate the normal mode frequencies, eigenvectors (draw them),

          /|  k    _   k    _    k       k   |/
          /|zzzzzz|m|zzzzzz|m|zzzzzz|m|zzzzzz|/        Symmetric
          /|       -        -                |/

b) Illustrate each mode with a cartoon or picture.

c) How many non-zero-matrix elements are there?  How many different ones are there
(i.e. how many independent matrix elements are ther for this case?)


4) Now imagine that the three masses in the above problem are connected in a ring
geometry (equilateral triangle), but can each only move in one dimension (away from the center).
a) how many independent elements does the matrix for this system have?
b) what is the form of that matrix?  what are its eigenvalues and eigenvectors?
c) do a comparitive analysis of this system and the one above (prob. 3).  What underlying
principle or symmetry "organizes" or "controls" the dynamics of this system?

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