Originally Posted by

**CaptainBlack**

3. Rewrite as a vector equation X''=AX', where A is now a 2x2 matrix, and

X=[x,y]'.

4. Now the normal modes correspond to the eigen vectors of A.

The system of ODE is:

which may be rewritten as:

.

So writing , this becomes:

,

where:

Now we may find the eigen values and vectors of by hand, but I will resort to numerical packages at this point:

Code:

>A=[-73.333,6.667;6,-80]
-73.333 6.667
6 -80
>help eigen
function eigen (A)
## eigen(A) returns the eigenvectors and a basis of eigenvectors of A.
## {l,x,ll}=eigen(A), where l is a vector of eigenvalues,
## x is a basis of eigenvectors,
## and ll is a vector of distinct eigenvalues.
>
>{aa,bb,cc}=eigen(A)
-69.5171+0i -83.8159+0i
>aa
-69.5171+0i -83.8159+0i
>bb
0.867895+0i -0.536649+0i
0.496748+0i 0.843805+0i
>cc
-69.5171+0i -83.8159+0i
>

So we have eigen values and with corresponding eigen vectors:

and .

RonL

I will complete the last part of this tommorow, its midnight here now and I

am going to bed.