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In the book introduction to classical Mechanics by Kleppner and Kolenkow, while dealing with the analysis of molecular vibrations in a poliatomic molecule, they propose the following method, in order to find the normal modes of the molecule:

Look for the special solutions of the form

$x_{i} =a_{i}\sin{(wt + \phi)}$

$i=1,...,N$

Where $N$ is the number of atoms in the molecule

The argument for supporting the method is that masses in the molecule had to had the same frequency $w$ and the same phase factor in order to linear momentum to conserve.

I understand that the velocity of each mass is proportional to this frequency since the motion presented is going to be periodic, but I can't really understand fully the argument of why frequencies of the masses can't vary.

Edit: After getting an answer for my doubt I realize that I did not state in a good way, what was my doubt. Here is the question that I intended to ask: For getting the normal modes we assume that all masses have the same frequency, But how can we assure the existence of that solution? In kleppner they say that it is because of linear momentum conservation, but I don't really get why.

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That is because we are looking for solutions of this kind. We are looking for solutions where all the masses are moving with the same frequency.

Why are we looking for only these? The system of masses can obviously be oscillating with arbitrary and varying frequency so why are we restricting ourselves to boring uniform solutions? Turns out, every possible motion of your system can be expressed in terms of these uniform motions, the ones we call normal modes. Clearly it is easier to describe the system in terms of normal modes than by trying to keep track of shat each mass is doing when, especially when the motion is very complicated.

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    $\begingroup$ Now I understand that was the solutions we are searching, but how can we assure that it exist? $\endgroup$
    – CristhianMartinez08
    Commented Jan 30, 2020 at 5:29
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    $\begingroup$ We solve the equations assuming solutions of this form. If we are unable to do so then it doesn’t exist. But in all cases we are interested in, they exist. This is because the equations are all second order linear differential equations where the second derivative of a function is proportional to the function itself. $\endgroup$
    – Superfast Jellyfish
    Commented Jan 30, 2020 at 5:33
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    $\begingroup$ thanks, so the strategy is based in the type of ODE, rather than the physical meaning? $\endgroup$
    – CristhianMartinez08
    Commented Jan 30, 2020 at 5:39
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    $\begingroup$ Yes! But physically speaking these normal modes are the ones in which total energy is minimised. This can be thought of as minimising repulsion on average. Thus most physical systems can be found in normal modes if we wait long enough for other motions to have decayed out. $\endgroup$
    – Superfast Jellyfish
    Commented Jan 30, 2020 at 5:42

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