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Normal modes of oscillator

  • Thread starter ismaili
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1. Homework Statement

I'm reading Landau's Mechanics, in section 23, he discusses the oscillations with more than one degree of freedom, the Lagrangian is

[tex]L = \frac{1}{2}\left(m_{ik}\dot{x}_i\dot{x}_k - k_{ik}x_ix_k\right)[/tex]

where [tex]m_{ik},k_{ik}[/tex] are symmetric constants, and the summation over [tex]i,k[/tex] in the above equation is understood.
By substituting the form of solutions

[tex] x_k = A_k\exp(i\omega t)[/tex]

we get the system of linear equations,

[tex]\sum_i\left(-\omega^2m_{ik} + k_{ik}\right)A_k = 0\quad\cdots(*)[/tex]

In order to have non-trivial solution, the determinant of the following matrix should be zero,

[tex]\left| k_{ik} - \omega^2m_{ik} \right| = 0\quad\cdots(**)[/tex]

The roots of [tex]\omega[/tex] are denoted as [tex]\omega_\alpha[/tex].
Then comes my question. He said that "The frequencies [tex]\omega_\alpha[/tex] having been found, we substitute each of them in eq(*) and find the corresponding coefficients [tex]A_k[/tex]. If all the roots [tex]\omega_\alpha[/tex] of the characteristic equation are different, the coefficients [tex]A_k[/tex] are proportional to the minors of the determinant (eq(**)) with [tex]\omega=\omega_\alpha[/tex]."

My question is the sentence with the underline, I think this is a problem of linear algebra actually, but I can't come up with any idea and can't find the material in Wiki.

Thanks for your solution or reference!
 

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