Mathematical Methods for Physicists: A concise introduction - Site Map

MATRIX ALGEBRA
distance between any particle pair constant at all times. Then its angular momentum
about the origin O of the coordinate system is
L ˆ X
m r v ˆ X
m r …x r †
where the subscript refers to mass m a located at r ˆ…x 1 ; x 2 ; x 3 †, and x the
angular velocity of the rigid body.
Expanding the vector triple product by using the vector identity
A …B C† ˆB…A C†C…A B†;
we obtain
L ˆ X
m br 2 x r …r x†c:
In terms of the components of the vectors r and x, the ith component of L i is
L i ˆ X " #
X 3
X
m ! i x 2 3
;k x ;i x ; j ! j
ˆ X X
! j
kˆ1
m
j
k
jˆ1
" #
X
ij x 2 ;k x ;i x ;j ˆ X
I ij ! j
j
or
~L ˆ ~I ~!:
Both ~L and ~! are three-dimensional column vectors, while ~I is a 3 3 matrix and
is called the moment inertia matrix.
In general, the angular momentum vector L of a rigid body is not always
parallel to its angular velocity x and ~I is not a diagonal matrix. But we can orient
the coordinate axes in space so that all the non-diagonal elements I ij …i 6ˆ j†
vanish. Such special directions are called the principal axes of inertia. If the
angular velocity is along one of these principal axes, the angular momentum
and the angular velocity will be parallel.
In many simple cases, especially when symmetry is present, the principal axes of
inertia can be found by inspection.
Normal modes of vibrations
Another good illustrative example of the application of matrix methods in classical
physics is the longitudinal vibrations of a classical model of a carbon dioxide
molecule that has the chemical structure O±C±O. In particular, it provides a good
example of the eigenvalues and eigenvectors of an asymmetric real matrix.
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