4.3 Applications 53
Here
(
r⊥(i)) 2
=
(
r(i)) 2
−
(
̂wνrν(i)) 2
is the square of the shortest distance of mass pointifrom the rotation axis.
The moment of inertia tensor and the inertia moment, as defined by (4.23) and
(4.24) depend on the choice of the origin. As mentioned before, the origin for the
position vectors has to be on the axis of rotation. When the axis of rotation goes
through the center of mass of the body, it is convenient to choose the center of mass
as the origin. In this case one has
∑N
i= 1 mir(i)=0.
Due to its definition, the moment of inertia tensor does not change its sign both
under the parity operation nor under time reversal.
When one talks aboutthemoment of inertia tensor orthemoment of inertia of a
solid body, one means quantities, which are characteristic for the shape of the body:
they are computed via (4.23) and (4.24) withr(i)=0 corresponding to the center
of mass of the solid body. Furthermore, expressions analogous to (4.23) and (4.24)
are presented later for continuous mass distributions, where the sum over discrete
masses is replaced by an integral over space.
Examples for the moment of inertia tensor are discussed in Sects.5.3.1,8.3.3and
computed in the Exercises5.1and8.4.
4.4 Dual Relation and Epsilon-Tensor in 2D
4.4.1 Definitions and Matrix Notation
Letaiandbi,i = 1 ,2, be the Cartesian components of two vectors in 2D. The
antisymmetric part of the dyadic constructed from these components is related to a
scalarcaccording to
c=a 1 b 2 −a 2 b 1. (4.25)This dual relation can be written as
c=εijaibj, (4.26)where the summation convention is used for the Latin subscripts. By analogy to the
3D case,εijis defined by
εij:=∣
∣
∣
∣
δ1iδ1j
δ2iδ2j