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Coupled Harmonic Oscillators. In this series of exercises you are asked

to generalize the material on harmonic oscillators in Section 6.2 to the

case where the oscillators are coupled. Suppose there are two masses m1

and m2 attached to springs and walls as shown in Figure 6.10. The springs

connecting mj to the walls both have spring constants k1, while the spring

connecting m1 and m2 has spring constant k2. This coupling means that

the motion of either mass affects the behavior of the other.

Let xj denote the displacement of each mass from its rest position, and

assume that both masses are equal to 1. The differential equations for

these coupled oscillators are then given by

x_{1}'' = -k(_{1}+ k_{2})x_{1}+ k_{2}x_{2}

x_{2}'' = k_{2}x_{1}- (k_{1}+ k_{2})x_{2}

These equations are derived as follows. If m1 is moved to the right

(x1 > 0), the left spring is stretched and exerts a restorative force on

m1 given by -k1x1. Meanwhile, the central spring is compressed, so it

exerts a restorative force on m1 given by -k2x1. If the right spring is

stretched, then the central spring is compressed and exerts a restorative

force on m1 given by k2x2 (since x2 < 0). The forces on m2 are similar.

(a) Write these equations as a first-order linear system.

(b) Determine the eigenvalues and eigenvectors of the corresponding

matrix.

(c) Find the general solution.

(d) Let ω1 = √k1 and ω2 = √k1 + 2k2. What can be said about the

periodicity of solutions relative to the ωj? Prove this.

2. Relevant equations

(A - λI) = Ax

3. The attempt at a solution

Part a is ok but I'm stuck for the rest

For part a we set y_{1}= dx_{1}/dt and therefore dy_{1}/dt = d^{2}x/dt^{2}

Then y_{2}= dx_{2}/dt so we have a similar case of dy_{2}/dt = d^{2}x/dt^{2}

I think this is enough to satisfy part a) of the question

So then we put them in to matrix form to try and find the eigenvalues and vectors and get the following by rows. Columns are x1,y1, x2, y2 respectively.

dx_{1}/dt = 0 1 0 0

dy_{1}/dt = -(k1+k2) 0 k2 0

dx_{2}/dt = 0 0 0 1

dy_{2}/dt = k2 0 -(k1+k2) 0

So there is a diagonal of zeros meaning we have zero trace. By using the formula (A - λI) = Ax. An attempt was made to try and get the eigenvalues. I subtract λ across each term in the diagonal, multiply out the expression and I end up with the following expression.

k1^{2}λ + k1k2λ + k2^{2}λ. I don't know how to deal with this expression. I'm used to seeing things in the form λ^{2}+ λ - c.

So any helpful tips?

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# Homework Help: Eigenvalues and vectors of a 4 by 4 matrix

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