Editing Dimensional analysis (section)

=== Siano's extension: orientational analysis ===
{{see also|Angle#Dimensional analysis}}

[[Angle]]s are, by convention, considered to be dimensionless quantities (although the wisdom of this is contested <ref>{{ cite journal  | title=Angles in the SI: a detailed proposal for solving the problem | year=2021 | pages=053002 | journal=Metrologia | doi=10.1088/1681-7575/ac023f | volume=58 | issue=5 | url=http://dx.doi.org/10.1088/1681-7575/ac023f | last1=Quincey | first1= Paul | arxiv=2108.05704 | bibcode=2021Metro..58e3002Q }}</ref>) . As an example, consider again the projectile problem in which a point mass is launched from the origin {{math|1=(''x'', ''y'') = (0, 0)}} at a speed {{math|''v''}} and angle {{math|''θ''}} above the ''x''-axis, with the force of gravity directed along the negative ''y''-axis. It is desired to find the range {{math|''R''}}, at which point the mass returns to the ''x''-axis. Conventional analysis will yield the dimensionless variable {{math|1=''π'' = ''R'' ''g''/''v''<sup>2</sup>}}, but offers no insight into the relationship between {{math|''R''}} and {{math|''θ''}}.

Siano has suggested that the directed dimensions of Huntley be replaced by using ''orientational symbols'' {{math|1<sub>x</sub>&nbsp;1<sub>y</sub>&nbsp;1<sub>z</sub>}} to denote vector directions, and an orientationless symbol 1<sub>0</sub>.<ref>{{harvs|txt=yes|last=Siano|year1=1985-I|year2=1985-II}}</ref> Thus, Huntley's L<sub>{{math|x}}</sub> becomes L1<sub>{{math|x}}</sub> with L specifying the dimension of length, and {{math|1<sub>x</sub>}} specifying the orientation. Siano further shows that the orientational symbols have an algebra of their own.  Along with the requirement that {{math|1=1<sub>''i''</sub><sup>−1</sup> = 1<sub>''i''</sub>}}, the following multiplication table for the orientation symbols results:

{| class="wikitable"
! !! <math>\mathbf{1_0}</math> !! <math>\mathbf{1_\text{x}}</math> !! <math>\mathbf{1_\text{y}}</math> !! <math>\mathbf{1_\text{z}}</math>
|-
! scope="col" | <math>\mathbf{1_0}</math>
| <math>1_0</math> || <math>1_\text{x}</math> || <math>1_\text{y}</math> || <math>1_\text{z} </math>
|-
! scope="col" | <math>\mathbf{1_\text{x}}</math>
| <math>1_\text{x}</math> || <math>1_0</math> || <math>1_\text{z} </math> || <math>1_\text{y}</math>
|-
! scope="col" | <math>\mathbf{1_\text{y}}</math>
| <math>1_\text{y}</math> || <math>1_\text{z} </math> || <math>1_0</math> || <math>1_\text{x}</math>
|-
! scope="col" | <math>\mathbf{1_\text{z}}</math>
| <math>1_\text{z} </math> || <math>1_\text{y}</math> || <math>1_\text{x}</math> || <math>1_0</math>
|}

The orientational symbols form a group (the [[Klein four-group]] or "Viergruppe"). In this system, scalars always have the same orientation as the identity element, independent of the "symmetry of the problem".  Physical quantities that are vectors have the orientation expected:  a force or a velocity in the z-direction has the orientation of {{math|1<sub>z</sub>}}.  For angles, consider an angle {{mvar|θ}} that lies in the z-plane.  Form a right triangle in the z-plane with {{mvar|θ}} being one of the acute angles.  The side of the right triangle adjacent to the angle then has an orientation {{math|1<sub>x</sub>}} and the side opposite has an orientation {{math|1<sub>y</sub>}}.  Since (using {{math|~}} to indicate orientational equivalence) {{math|1=tan(''θ'') = ''θ''&nbsp;+&nbsp;... ~ 1<sub>y</sub>/1<sub>x</sub>}} we conclude that an angle in the xy-plane must have an orientation {{math|1=1<sub>y</sub>/1<sub>x</sub> = 1<sub>z</sub>}}, which is not unreasonable.  Analogous reasoning forces the conclusion that {{math|1=sin(''θ'')}} has orientation {{math|1<sub>z</sub>}} while {{math|cos(''θ'')}} has orientation 1<sub>0</sub>.  These are different, so one concludes (correctly), for example, that there are no solutions of physical equations that are of the form {{math|''a'' cos(''θ'') + ''b'' sin(''θ'')}}, where {{mvar|a}} and {{mvar|b}} are real scalars. An expression such as <math>\sin(\theta+\pi/2)=\cos(\theta)</math> is not dimensionally inconsistent since it is a special case of the sum of angles formula and should properly be written:
: <math>
  \sin\left(a\,1_\text{z} + b\,1_\text{z}\right) =
    \sin\left(a\,1_\text{z}) \cos(b\,1_\text{z}\right) +
    \sin\left(b\,1_\text{z}) \cos(a\,1_\text{z}\right),
</math>
which for <math>a = \theta</math> and <math>b = \pi/2</math> yields {{tmath|1=\sin(\theta\,1_\text{z} + [\pi/2]\,1_\text{z}) = 1_\text{z}\cos(\theta\,1_\text{z})}}. Siano distinguishes between geometric angles, which have an orientation in 3-dimensional space, and phase angles associated with time-based oscillations, which have no spatial orientation, i.e. the orientation of a phase angle is {{tmath|1_0}}.

The assignment of orientational symbols to physical quantities and the requirement that physical equations be orientationally homogeneous can actually be used in a way that is similar to dimensional analysis to derive more information about acceptable solutions of physical problems.  In this approach, one solves the dimensional equation as far as one can.  If the lowest power of a physical variable is fractional, both sides of the solution is raised to a power such that all powers are integral, putting it into [[Canonical form|normal form]].  The orientational equation is then solved to give a more restrictive condition on the unknown powers of the orientational symbols. The solution is then more complete than the one that dimensional analysis alone gives. Often, the added information is that one of the powers of a certain variable is even or odd.

As an example, for the projectile problem, using orientational symbols, {{math|''θ''}}, being in the xy-plane will thus have dimension {{math|1<sub>z</sub>}} and the range of the projectile {{mvar|R}} will be of the form:
: <math>R = g^a\,v^b\,\theta^c\text{ which means }\mathsf{L}\,1_\mathrm{x} \sim
\left(\frac{\mathsf{L}\,1_\text{y}}{\mathsf{T}^2}\right)^a \left(\frac{\mathsf{L}}{\mathsf{T}}\right)^b\,1_\mathsf{z}^c.\,</math>

Dimensional homogeneity will now correctly yield {{math|1=''a'' = −1}} and {{math|1=''b'' = 2}}, and orientational homogeneity requires that {{tmath|1=1_x /(1_y^a 1_z^c)=1_z^{c+1} = 1}}. In other words, that {{mvar|c}} must be an odd integer. In fact, the required function of theta will be {{math|sin(''θ'')cos(''θ'')}} which is a series consisting of odd powers of {{mvar|θ}}.

It is seen that the Taylor series of {{math|sin(''θ'')}} and {{math|cos(''θ'')}} are orientationally homogeneous using the above multiplication table, while expressions like {{math|cos(''θ'') + sin(''θ'')}} and {{math|exp(''θ'')}} are not, and are (correctly) deemed unphysical.

Siano's orientational analysis is compatible with the conventional conception of angular quantities as being dimensionless, and within orientational analysis, the [[radian]] may still be considered a dimensionless unit. The orientational analysis of a quantity equation is carried out separately from the ordinary dimensional analysis, yielding information that supplements the dimensional analysis.