?It is a crucial property of the eigenvectors, \(\mathbf{a}_{(1)},\cdots,\mathbf{\

QUESTION:

It is a crucial property of the eigenvectors, \(\mathbf{a}_{(1)},\cdots,\mathbf{\ a}_{(n)}\), describing the motion in the normal modes of an oscillating system that any \((n \times 1)\) column x can be written as a linear combination of the n eigenvectors; that is, the eigenvectors are a basis of the space of \((n \times 1)\) columns. This is proved in the appendix, but to illustrate it, do the following: (a) Write down the three eigenvectors \(\mathbf{a}_{(1)},\mathbf{\ a}_{(2)}\), and \(\mathbf{a}_{(3)}\) for the coupled pendulums of Section 11.6. (Each of these contains an arbitrary overall factor, which you can choose at your convenience.) Prove that they have the property that any \((3 \times 1)\) column x can be expanded in terms of them. (b) The expansion coefficients in this expansion are the normal coordinates \(\xi_1,\ \xi_2,\ \xi_3\). Find the normal coordinates for the three coupled pendulums, and explain the sense in which they describe the three normal modes of Figure 11.14.

Text Transcription:

a_(1), cdot, a_(n)

(n times 1)

a_(1), a_(2)

a_(3)

(3 times 1)

xi_1, xi_2, xi_3