Regarding John Mallinckrodt's "elaborate and absurdly pedantic" method of
treating "angle" as a base quantity when doing dimensional power counting
(and introducing "a parameter called r' which is the 'tangential distance
per unit angle'" as elaborated on in his post of 12 SEP 96), I've been
inspired to describe the following:
Even though I am loath to actually advocate that "angle" be considered a
basic dimensional quantity rather than to just consider angles as being
dimensionless as per normal tradition, John's reiterating of his method
has inspired me to try to creatively consider alternative methods of
including angles as having a fundamental base dimension (let's call it A)
of their own. This led me to *tenatively* propose the following
alternate (even *more absurdly pedantic*) scheme (which I haven't yet
explored in detail to check for hidden bugs). Since I don't really like
the idea of 'angle' as a dimensioned quantity, what is described below is
not a serious proposal (in case anyone was wondering).
Consider the fully antisymetric 3-index Levi-Civita [epsilon] symbol that
is often introduced in 3 dimensions to facilitate the taking of cross
products between vectors. (Actually, what it really does is facilitate
the mapping of bivectors or 2-forms in 3 dimensions to so-called dual
'axial vectors' or 'pseudo-vectors'.) This 3-index symbol is an example
of a 'tensor density' and is often called a pseudo-tensor because of its
unconventional behavior under spatial inversions. Suppose we represent
3-vectors (and other 3-tensors) by their Cartesian coordinates with a
generic index for each basis vector. I.e., we let A_i mean the i-th
component of the vector [bold] A where the index i runs over the
Cartesian directions x,y, & z. Then in this case we can represent the
vector cross product C = A x B as below in terms of their coordinates
while using the Einstein summation convention and ignoring any
distinction between covariant and contravariant indices since we use only
Cartesian coordinates in our Euclidean 3-space (no curvilinear
coordinates, no curvature, no unconventionally signatured metric). We
use Latin letters from the middle of the alphabet as vector indices. Then
we have: C_k = A_i*B_j*[epsilon]_i_j_k. Here the value of
[epsilon]_i_j_k is traditionally taken to be the pure number +1 if i, j
& k are an even permutation of x, y & z, the pure number -1 if i, j & k
are an odd permutation of x,y & z, and the number 0 if i, j & k have any
of their values in common with each other (x,x,y or z,z,z for instance).
My proposal for including 'angle' as a base dimension is to arbitrarily
consider the Levi-Civita [epsilon] symbol to have the dimension of
inverse 'angle', (A^-1) while keeping its numerical values the same as
the traditional ones (as long as we agree to measure all angles in
radians). Thus the nonzero values of the epsilon symbol are to be
+/- 1 rad^-1. *But* if we decide to measure our angles in degrees then
the [epsilon] symbol is to have its values as +/- [pi]/180 deg^-1. This
change in the dimension of the [epsilon] symbol would have the effect of
giving the cross product of two vectors a *different* dimension than
their dot product. The dot product of two vectors would still have the
dimension of the product of the dimensions of its factors, but the cross
product of two vectors would have this dimension *divided by angle*. The
result of this change would give the angular momentum vector the
dimension M*L^2*T^-1*A^-1. The dimension of torque would then be
M*L^2*T^-2*A^-1. And the dimension of the rotational inertia tensor or
its projected scalar moment of inertia would have the dimension of
M*L^2*A^-2. This would give these quantities different dimensions than
action, work and mass quadrupole moment respectively which would still be
missing the corresponding inverse factors of the angle dimension. In the
new system we still have [work] = [torque]*[angle], [action] =
[angular momentum]*[angle], and [rotational kinetic energy] =
(1/2)*[moment of inertia]*[angular velocity]^2, and [angular momentum] =
[moment of inertia]*[angular velocity], but now the radians (or any other
angular unit we desire to use) are *not* dropped when doing these
multiplications. They stay present and cancel against the inverse angle
dimensions coming from the Levi-Civita [epsilon] symbol.
If we are comfortable with using the SI system we shouldn't get too upset
about arbitarily giving the [epsilon] symbol an inverse angle dimension
so that its usual values are in inverse radians when deciding to consider
'angle' as a basic dimensioned quantity. Recall that the SI system
arbitrarily defines a constant called [mu]_0 to have the dimension
[force]/[electric current]^2 so that [mu]_0 = 4*[pi]*10^-7 N/A^2 when its
inventors decided to consider 'electric current' as a basic dimensioned
quantity. Also, they arbitrarily define the Avogadro constant N_A to
have the dimension 1/[amount of substance] so that
N_A = 6.02214199(47) x 10^23 mol^-1 when they decided to make 'amount of
substance' a basic dimensioned quantity.