Tensors for Physics

30 2Basics

1-direction. Forw>0, the motion runs counterclockwise, i.e. in the mathemat-

ically positive sense. AssumingR=const.andw=const., the components of

the velocity are

v 1 =−Rwsin(wt), v 2 =Rwcos(wt), v 3 = 0.

In this case, the velocity is perpendicular to the position vector, it is purely tan-

gential.

2.7.3 Radial and Azimuthal Components of the Velocity

The position vectorrμ(t)can be written as a product of its magnituder=(rνrν)^1 /^2
and the unit vector̂rμ(t), according torμ=r̂rμ. Then one has

d

dt

rμ=

dr

dt

̂rμ+r

d

dt

̂rμ. (2.63)

Theradialcomponent of the velocity is the first term on the right hand side of (2.63),
which is parallel to the position vectorr. It describes the change of the length ofr.The
second term, associated with the change of the direction ofr, is called theazimuthal
component, sometimes also thetangential componentof the velocity, because it is
perpendicular tor. This can be seen quickly as follows. Notice that̂rν̂rν=1. The
time derivative of this equation yields 2̂rνddt̂rν=0, which implies that the derivative
of the radial unit vector̂ris perpendicular tôr. Alternatively, the definition of the unit
vector, viz.,̂rμ=rμr−^1 =rμ(rνrν)−^1 /^2 and the chain rule can be used to obtain

d

dt

̂rμ=r−^1

d

dt

rμ−r−^3 rμrν

d

dt

rν=r−^1 (δμν−̂rμ̂rν)vν. (2.64)

The projection tensorδμν−̂rμ̂rνguarantees thatddt̂rμis perpendicular tôrμ.
Notice that the wordtangentialis used with two slightly different meanings,
which only coincide for the motion on a circle. In one case it refers to the tangent
of a trajectory which points in the direction of the velocity. In the second case,
just discussed here, where the word “azimuthal” is more appropriate, it means the
direction perpendicular to the position vector.

2.8 Time Reversal

The trajectory of a particle or of the center of mass of an extended object is described
by the time dependence of the position vectorr=r(t). One may ask the question:
does the trajectoryr(−t)also describe a physically possible motion? In other words,