44 3 Symmetry of Second Rank Tensors, Cross Product
Fig. 3.2 Motion on
a straight line
The time change of the orbital angular momentumLisL ̇=r ̇×p+r×p ̇. Notice
thatr ̇=v. When one hasp=mv,thetermr ̇×pis zero. Then the time change of
the orbital angular momentum is given by
dL
dt:=L ̇=r×p ̇=r×F. (3.45)The quantityr×Fis calledtorque.
Notice: there are two cases whereL ̇=0 and where, as a consequence, the angular
momentumLis constant:
- no force is acting,F=0,
- the forceFis parallel to the position vectorr.
A force with this property is calledcentral force.
Furthermore, notice: the orbital angular momentum and the torque are axial vec-
tors. These quantities do not change sign upon the parity operation. This follows from
the fact that they are bilinear functions of polar vectors. By definition, the angular
momentum changes sign under time reversal, its time derivative does not change
sign. Thus (3.45) describes a reversible dynamics provided that the torquer×F
does not change sign under time reversal. This, in turn, is the case when there is no
rotational friction proportional to the rotational velocity.
3.4 Exercise: Torque Acting on an Anisotropic Harmonic Oscillator
Determine the torque for the force
F=−kr·ee−(r−er·e),where the parameterkand unit vectoreare constant. Which component of the angular
momentum is constant, even fork=1?
3.4.3 Motion on a Circle
The velocityvof a mass point on a circular orbit can be expressed as
v=w×r, (3.46)