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.

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.

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

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.