It is not dicult to verify that 1 cosjn 1 cos2jn

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Unformatted text preview: the symmetric matrix given above. To find the frequencies of vibration of our mechanical system, we need to find the eigenvalues of the n × n matrix A. To simplify the calculation of the eigenvalues of this matrix, we make use of the fact that the carts are identically situated—if we relabel the carts, shifting them to the right by one, the dynamical system remains unchanged. Indeed, lew us define new coordinates (y1 , . . . , yn ) by setting x1 = y2 , x2 = y3 , ... , xn−1 = yn , xn = y1 , or in matrix terms, x = T y, where T = 0 0 0 · 0 1 1 0 0 · 0 0 0 1 0 · 0 0 ··· ··· ··· ··· ··· ··· 0 0 0 · 1 0 . Then y satisfies exactly the same system of differential equations as x: d2 y k = Ay. 2 dt m 55 (2.18) On the other hand, d2 x k = Ax 2 dt m ⇒ T d2 y k = AT y 2 dt m d2 y k = T −1 AT y. 2 dt m or Comparison with (2.18) yields A = T −1 AT, or T A = AT. In other words the matrices A and T commute. Now it is quite easy to solve the eigenvector-eigenvalue problem for T . Indeed, if x is an eigenvecto...
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This document was uploaded on 01/12/2014.

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