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I solved the problem of small oscillations for a 3-atom molecule, such as CO_{2}, which is modeled as 3 masses connected by 2 springs. Both springs have a constant k, the outer masses are m and the middle one is M.

There are 3 modes of oscillations, and one of them is of course [tex]\omega[/tex] = 0, i.e it is a rigid translation of the molecule as a whole. I've also found the normal coordinates for each mode, and for this particular one I found:

[tex]Q = \frac{1}{\sqrt{2m+M}} (mq_1 + Mq_2 + mq_3)[/tex]

Where q_{i}is the "real" coordinates of each molecule. This seems pretty logical, right? Because basically I foud that the normal coordinates for the [tex]\omega[/tex] = 0 mode is exactly the coordinate of the center of mass (after normalization).

However, the exact same problem was solved in Goldstein's "Classical mechanics" (3rd ed.), and a different normal coordinate was found there. It was:

[tex]Q = \frac{1}{\sqrt{2m+M}} (\sqrt{m}q_1 + \sqrt{M}q_2 + \sqrt{m}q_3)[/tex]

Which is not what I found, nor do I understand its meaning. My friend thinks that both answers are correct, and the difference is just in normalization; I don't agree, one of these answers must be wrong. I'd think that my answer is correct, but since the other one is taken for the book, I'm not so sure.

Can someone please clarify? Which answer seems more logical?

Thanks,

Chen

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# Homework Help: Normal coordinates (small oscillations)

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