Question 1: Why is it that the eigenvalues of L^x and L^y are ±1? (Hint: Consider L^x o L^x and L^y o L^y.)
Question 2: Show that L^x and L^y commute.
Question 3: The result of Part (a) implies that we can use L^x to divide the normal modes of the same frequency into 2 classes, one is symmetric under the operation of L^x, i.e. L^x ψ = ψ, and the other which is antisymmetric under the operation of L^x, i.e., L^xψ = -ψ. Similarly, one can say the same with L^y. Using the result of Part (b), we thus arrive at the conclusion that the normal modes can be divided into four classes, each possessing a definite symmetry under the operation of either L^x or L^y. In this part through Part (e) we will work out explicitly the class which satisfies L^xψ = ψ and L^yψ = ψ. Explain why the solutions in this class must be of form ψ = (ξ, η, η, ξ, η, η).
Question 4: Referring to the figure above, explain why the equations of motion for Balls A and B are, respectively,
M d2ξ/dt2 = -k1(ξ- ξ) - k2(ξ - η) - k2(ξ - η) = -2k2(ξ-η),
m d2η/dt2 = -k2(η - ξ).
Question 5: Find the normal mode frequencies and their associated normal modes (ξ, η) of the equations of Part (d).
Question 6: Write down the equations of motion for the class of solutions satisfying L^xψ = ψ and L^y ψ = - ψ.
Question 7: Find the normal mode frequencies and their associated normal modes (ξ, η) of the equations of Part (f).
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