7—Operators and Matrices 178The integrals in Eq. ( 14 ) are simply sums this time, and the sums have only two terms. I’m making the
approximation that these are point masses. Make the coordinate system match the indicated basis, withxright
andyup, thenzis zero for all terms in the sum, and the rest are
∫
dm(y^2 +z^2 ) =m 1 r 12 cos^2 α+m 2 r^22 cos^2 α
−
∫
dmxy=−m 1 r^21 cosαsinα−m 2 r 22 cosαsinα
∫
dm(x^2 +z^2 ) =m 1 r 12 sin^2 α+m 2 r^22 sin^2 α
∫
dm(x^2 +y^2 ) =m 1 r 12 +m 2 r^22The matrix is then
(I) =(
m 1 r^21 +m 2 r^22)
cos^2 α −cosαsinα 0
−cosαsinα sin^2 α 0
0 0 1
(15)
Don’t count on all such results factoring so nicely.
In this basis, the angular velocity~ωhas only one component, so what is~L?
(
m 1 r^21 +m 2 r 22)
cos^2 α −cosαsinα 0
−cosαsinα sin^2 α 0
0 0 1
0
ω
0
=
(
m 1 r^21 +m 2 r^22)
−ωcosαsinα
ωsin^2 α
0
Translate this into vector form:
~L=(
m 1 r^21 +m 2 r 22)
ωsinα(
−~e 1 cosα+~e 2 sinα)
(16)
Whenα= 90◦, thencosα= 0and the angular momentum points along they-axis. This is the symmetric special
case where everything lines up along one axis. Notice that ifα= 0then everything vanishes, but then the masses