# Harmonic Oscillators: Differential Equations

The book being used for this course is Differential Equations, Dynamical Systems, and an Introduction to Chaos by Morris W. Hirsch.

The question is as follows.

Suppose there are two masses $m_1$ and $m_2$ attached to springs and walls. The springs connecting $m_j$ to the walls both have spring constants $k_1$, while the springs connecting $m_1$ and $m_2$ has spring constant $k_2$. This coupling means that the motion of either mass affects the behavior of the other.

Let $x_j$ denote the displacement of each mass from its rest position, and assume that both masses are equal to 1. The differential equation for these coupled oscillators are then given by \begin{eqnarray} x_1^{"}& = & -(k_1+k_2)x_1+k_2x_2\\ x_2^{"}& = & k_2x_1-(k_1+k_2)x_2\\ \end{eqnarray}

• Write these equations as a first-order linear system.
• Determine the eigenvalues and eigenvectors of the corresponding matrix.
• Find the general solution.
• Let $\omega_1= \sqrt{k_1}$ and $\omega_2=\sqrt{k_1+2k_2}$. What can be said about the periodicity of solutions relative to the $\omega_j$? Prove this.

Now I just want to focus on the first part: to see if my linear system is correct. I first start by introducing the new variable $y_j=x^{'}_j$ for $j = 1,2$, so that the equations can be written as a system. \begin{eqnarray} x^{'}_1 & =& y_1\\ y^{'}_1 & =& -(k_1+k_2)x_1+k_2x_2\\ x^{'}_2 & =& y_2\\ y^{'}_2 & =& k_2x_1-(k_1+k_2)x_2 \end{eqnarray} In matrix form, this system is $X^{'}=AX$ where $X=(x_1,y_1,x_2,y_2)$ and \begin{align} A & = & \begin{bmatrix} 0 & 1&0 &0 \\ -(k_1+k_2)& 0& k_2& 0\\ 0& 0& 0& 1\\ k_2& 0& -(k_1+k_2)&0\\ \end{bmatrix} \end{align}

Do I have the right linear system?

$$x_1^\prime = [0, 1, 0, 0]\left[\begin{array}{c}x_1\\y_1\\x_2\\y_2 \end{array}\right] = y_1$$Which is correct, by your definition of $y_1$. The second row reads
$$x_1^{\prime\prime} = y_1^\prime = [-(k_1+k_2), 0, k_2, 0]\left[\begin{array}{c}x_1\\y_1\\x_2\\y_2 \end{array}\right] = -(k_1+k_2)x_1 + k_2x_2$$ which is the first of your original equations. Do this for rows 3 and 4 to finish checking your answer.