Consider 3 equal masses of mass m connected by equal springs with spring constant k, with...

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Consider 3 equal masses of mass m connected by equal springs with spring constant k, with masses 1 & 3 also connected to the walls by springs with constant k (diagram (c) from last week's homework, or Georgi 3.3 with all masses, springs the same). a) Write down the M¹K matrix (you did this last week). Write down S, the 3x3 reflection symmetry matrix. Show that M-¹KS=SM-¹K. b) What are the eigenvalues of S? Do not directly compute the eignevalues using the determinant for- mula, but instead use the fact that two mirror reflections is the same as the original. c) Find 3 eigenvectors of S. Show that the eigenvector with a unique eigenvalue is also an eigenvector of M-¹K. d) Unless you are very lucky in your initial guess, the two eigenvectors of S you found that have the same eigenvalue will not be eigenvectors of M¹K. Show that the two remaining eigenvectors of M-¹K you found last week are eigenvectors of S and show that these can be written as a linear combination of the eigenvectors you found already. Consider 3 equal masses of mass m connected by equal springs with spring constant k, with masses 1 & 3 also connected to the walls by springs with constant k (diagram (c) from last week's homework, or Georgi 3.3 with all masses, springs the same). a) Write down the M¹K matrix (you did this last week). Write down S, the 3x3 reflection symmetry matrix. Show that M-¹KS=SM-¹K. b) What are the eigenvalues of S? Do not directly compute the eignevalues using the determinant for- mula, but instead use the fact that two mirror reflections is the same as the original. c) Find 3 eigenvectors of S. Show that the eigenvector with a unique eigenvalue is also an eigenvector of M-¹K. d) Unless you are very lucky in your initial guess, the two eigenvectors of S you found that have the same eigenvalue will not be eigenvectors of M¹K. Show that the two remaining eigenvectors of M-¹K you found last week are eigenvectors of S and show that these can be written as a linear combination of the eigenvectors you found already.

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Solution 192 Given tha... View the full answer