$$ V(x_0, x_1)=\frac{1}{2}K(x_0-x_1)^2 $$

$$ F_0=-\frac{\partial}{\partial x_0}V(x_0, x_1)=-K(x_0-x_1) $$

$$ F_1=-\frac{\partial}{\partial x_1}V(x_0, x_1)=K(x_0-x_1) $$

$$ \omega_0=\sqrt\frac{K}{m_0} ===== 運動方程式 ===== $$ m_0\frac{d^2}{dt^2}x_0(t)=F_0=-K(x_0(t)-x_1(t)) $$ $$ m_1\frac{d^2}{dt^2}x_1(t)=F_1=K(x_0(t)-x_1(t)) $$ ===== フーリエ展開 ===== $$ x_0(t) = X_0(\omega)e^{-i\omega t} $$ $$ x_1(t) = X_1(\omega)e^{-i\omega t} $$ ---- $$ m_0\frac{d^2}{dt^2}x_0(t)=-m_0\omega^2X_0(\omega)e^{-i\omega t}=-K(X_0(\omega)e^{-i\omega t}-X_1(\omega)e^{-i\omega t})) $$ $$ m_1\frac{d^2}{dt^2}x_1(t)=-m_1\omega^2X_1(\omega)e^{-i\omega t}=K(X_0(\omega)e^{-i\omega t}-X_1(\omega)e^{-i\omega t})) $$ ===== 固有方程式 ===== $$ -m_0\omega^2X_0(\omega)=-KX_0(\omega)+KX_1(\omega) $$ $$ -m_1\omega^2X_1(\omega)=KX_0(\omega)-KX_1(\omega) $$ ---- $$ (K-m_0\omega^2)X_0(\omega)-KX_1(\omega)=0 $$ $$ -KX_0(\omega)+(K-m_1\omega^2)X_1(\omega)=0 $$ ---- $$ \begin{bmatrix} K-m_0\omega^2 & -K
-K & K-m_1\omega^2 \end{bmatrix} \begin{bmatrix} X_0(\omega)
X_1(\omega) \end{bmatrix}=O

$$