Toronto Math Forum
MAT2442013S => MAT244 MathLectures => Ch 3 => Topic started by: Victor Ivrii on January 31, 2013, 05:50:11 PM

Consider two identical and connected harmonic oscillators:
\begin{equation}
\left\{\begin{aligned}
&y''+K y + L (yz)=0\\
&z''+Kz + L(zy)=0
\end{aligned}\right.
\label{eq1}
\end{equation}
with $K>0$, $L>0$.
Even if this is a system one can add or subtract equations getting system of two equations describing $y+z$ and $yz$ separately.
1) Find $y+z$ and $yz$ and then $y$ and $z$ (so, find the general solution of (\ef{eq1}).
2) What frequencies has the described system?

2)
guess $ y = A e^{rt}$, $z = B e^{rt}$
$$
A r^2 + K A + L\left(AB\right) = 0 \\
B r^2 + K B + L\left(BA\right) = 0 \\
\left(
\begin{array}{cc}
r^2 + K + L &  L \\
L & r^2 + K + L \\
\end{array} \right)
\left( \begin{array}{c}
A \\
B
\end{array}
\right) = 0
$$
nontrivial solution exists if there is no inverse to the ugly matrix, therefore its determinant is 0
$$
\left( r^2 + K + L \right)^2 + L^2 = 0
$$
expand and use quadratic equation
$$
r^2 = K  L \pm L
$$
therefore the frequencies are $\pm K$, $\pm\left(K2L\right)$

Add and subtract the first and second equations to obtain:
\begin{align}
(y+z)'' + K(y+z) &= 0 \label{added} \\
(yz)'' + (K+2L)(yz) &= 0 \label{subtracted}
\end{align}
Since $K, L > 0$, both equations are of the form $u'' + \omega^2 u = 0$, with general solution $u = A \cos (\omega t) + B \sin (\omega t)$. So the general solution to $(\ref{added})$ is
\begin{equation}
y+z = A \cos (\sqrt{K} t) + B \sin (\sqrt{K} t)
\end{equation}
and the general solution to $(\ref{subtracted})$ is
\begin{equation}
y  z = C \cos (\sqrt{K+2L} t) + D \sin(\sqrt{K+2L}t)
\end{equation}
Using the identities $y = \frac{1}{2}((y+z)+(yz))$ and $z = \frac{1}{2}((y+z)(yz))$ we obtain the general solution to $(\ref{eq1})$:
\begin{align}
y &= A' \cos(\omega_1 t) + B' \sin(\omega_1 t) + C' \cos(\omega_2 t) + D' \sin(\omega_2 t) \\
z &= A' \cos(\omega_1 t) + B' \sin(\omega_1 t)  C' \cos(\omega_2 t)  D' \sin(\omega_2 t)
\end{align}
where $A' = A/2$ and so on, and the frequencies are $\omega_1 = \sqrt{K}$, and $\omega_2 = \sqrt{K+2L}$.