Note on the normalization of the normal mode eigenvectors.
The normalization constant is a little tricky, owing to the fact that we don’t have the traditional
secular equation. If the problem were
rrr
r
Va = λa ,
the normalization would be simply
r r
a ⋅a =1
or
∑a
2
i
=1
i
But here we have the more complicated case of
rrr
rrr
Va = λTa
Goldstein, in the text Classical Mechanics, chapter 10, shows that the generalized
normalization is
r rr r
a ⋅T ⋅ a = 1
or
∑ Tij a ik a jk = 1
i,j
For example, the first eigenvector becomes
1
 
1
2m + M  
1
1
With
rr  m 0
T =0 M
0 0

0

0
m 
you can readily see that the normalization criterion is satisfied.