How Does Cramer's Rule Apply to Finding Normal Modes in Oscillatory Systems?

[ismaili]

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

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

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

$$x_k = A_k\exp(i\omega t)$$

we get the system of linear equations,

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

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

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

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

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!

 

[NateD]

Homework EquationsThe equation is given in the statement.The Attempt at a SolutionI think the author is talking about the Cramer's rule for solving a system of linear equations. The Cramer's rule is a method of solving linear equations using determinants. For a system of linear equations, a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = b_1a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n = b_2...a_{n1}x_1 + a_{n2}x_2 + ... + a_{nn}x_n = b_nwhere a_{ij} are constants, the Cramer's rule states that the solution of the system is x_i = \frac{\left|\begin{array}{ccc}b_1 & a_{12} & \cdots & a_{1n} \\b_2 & a_{22} & \cdots & a_{2n} \\\vdots & \vdots & \ddots & \vdots \\b_n & a_{n2} & \cdots & a_{nn}\end{array}\right|}{\left|\begin{array}{ccc}a_{11} & a_{12} & \cdots & a_{1n} \\a_{21} & a_{22} & \cdots & a_{2n} \\\vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & \cdots & a_{nn}\end{array}\right|}.In our case, the matrix (**) is of the form \left| \begin{array}{ccc} a_{11}-\omega_\alpha^2m_{11} & a_{12}-\omega_\alpha^2m_{12} & \cdots & a_{1n}-\omega_\alpha^2m_{1n}\\a_{21}-\omega_\alpha^2m_{21