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Axial Vectors in Rotating Coordinate Systems

Axial vectors are vector cross products of ordinary position vectors. For example, angular momentum L=r×v and torque T=r×F are axial vectors.

Note that in the following the typographically distinction between vectors and scalars is that vectors are shown in color. Ordinary vectors are shown in red and axial vectors in green. Sometimes axial vectors are called pseudovectors and ordinary position vectors are called polar vectors. The significant difference between axial vectors and polar vectors is the effect on their coordinates of an inversion of the coordinate system; i.e.,

Under this inversion an ordinary (polar) vector P is transformed into its negative −P, whereas the coordinates of an axial vector A are unchanged by the inversion; i.e.,

The time derivative of a polar vector can be represented by the following formula

dP/dt = drP/dt + Ω×P

where dP/dt is the time derivative with respect to coordinates in an intertial frame of reference whereas drP/dt is the time derivative with respect to a frame of reference which is rotating about the vector Ω at a rate of rotation of Ω.

Now consider the axial vector A=B×C. Its time rate of change with respect to an inertial frame of reference is given by:

dA/dt = (dB/dt)×C + B×(dC/dt)

But dB/dt and dC/dt are given by:

dB/dt = drB/dt + Ω×B
and
dC/dt = drC/dt + Ω×C

This means that

dA/dt = (drB/dt + Ω×BC + B×(drC/dt + Ω×C)
which is equivalent to
dA/dt = (drB/dt)×C + B×(drC/dt)
+ (Ω×BC + B×(Ω×C)

The first two terms on the right are just drA/dt. The Lagrange formula for triple vector products shows that

(Ω×BC = -C×(Ω×B) = -[(C.B)Ω - (C.Ω)B]
and
B×(Ω×C) = (B.C)Ω - (B.Ω)C

When these expressions are substituted into the formula for dA/dt two terms cancel out leaving

dA/dt = drA/dt + (C.Ω)B - (B.Ω)C

When the above expression is rearranged to

dA/dt = drA/dt + (Ω.C)B - (Ω.B)C

the Lagrange triple vector product formula can be applied in reverse to yield

dA/dt = drA/dt + Ω×(B×C)

This shows that the formula for the time derivative of a vector under a rotating frame of reference applies to axial vectors as well as ordinary position vectors. It would also apply to hybrid vectors such as a vector cross product of a postion vector and an axial vector or the vector cross product of axial vectors.


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