Editing Dimensional analysis (section)

== Geometry: position vs. displacement ==

=== Affine quantities ===
{{further|Affine space}}
Some discussions of dimensional analysis implicitly describe all quantities as mathematical vectors. In mathematics scalars are considered a special case of vectors;{{citation needed|date=September 2013}} vectors can be added to or subtracted from other vectors, and, inter alia, multiplied or divided by scalars. If a vector is used to define a position, this assumes an implicit point of reference: an [[origin (mathematics)|origin]]. While this is useful and often perfectly adequate, allowing many important errors to be caught, it can fail to model certain aspects of physics. A more rigorous approach requires distinguishing between position and displacement (or moment in time versus duration, or absolute temperature versus temperature change).

Consider points on a line, each with a position with respect to a given origin, and distances among them. Positions and displacements all have units of length, but their meaning is not interchangeable:
* adding two displacements should yield a new displacement (walking ten paces then twenty paces gets you thirty paces forward),
* adding a displacement to a position should yield a new position (walking one block down the street from an intersection gets you to the next intersection),
* subtracting two positions should yield a displacement,
* but one may ''not'' add two positions.
This illustrates the subtle distinction between ''affine'' quantities (ones modeled by an [[affine space]], such as position) and ''vector'' quantities (ones modeled by a [[vector space]], such as displacement).
* Vector quantities may be added to each other, yielding a new vector quantity, and a vector quantity may be added to a suitable affine quantity (a vector space ''[[Group action (mathematics)|acts on]]'' an affine space), yielding a new affine quantity.
* Affine quantities cannot be added, but may be subtracted, yielding ''relative'' quantities which are vectors, and these ''relative differences'' may then be added to each other or to an affine quantity.

Properly then, positions have dimension of ''affine'' length, while displacements have dimension of ''vector'' length. To assign a number to an ''affine'' unit, one must not only choose a unit of measurement, but also a [[Origin (mathematics)|point of reference]], while to assign a number to a ''vector'' unit only requires a unit of measurement.

Thus some physical quantities are better modeled by vectorial quantities while others tend to require affine representation, and the distinction is reflected in their dimensional analysis.

This distinction is particularly important in the case of temperature, for which the numeric value of [[absolute zero]] is not the origin 0 in some scales. For absolute zero,
: −273.15&nbsp;°C ≘ 0&nbsp;K = 0&nbsp;°R ≘ −459.67&nbsp;°F,
where the symbol ≘ means ''corresponds to'', since although these values on the respective temperature scales correspond, they represent distinct quantities in the same way that the distances from distinct starting points to the same end point are distinct quantities, and cannot in general be equated.

For temperature differences,
: 1&nbsp;K = 1&nbsp;°C ≠ 1&nbsp;°F = 1&nbsp;°R.
(Here °R refers to the [[Rankine scale]], not the [[Réaumur scale]]).
Unit conversion for temperature differences is simply a matter of multiplying by, e.g., 1&nbsp;°F / 1&nbsp;K (although the ratio is not a constant value). But because some of these scales have origins that do not correspond to absolute zero, conversion from one temperature scale to another requires accounting for that. As a result, simple dimensional analysis can lead to errors if it is ambiguous whether 1&nbsp;K means the absolute temperature equal to −272.15&nbsp;°C, or the temperature difference equal to 1&nbsp;°C.

=== Orientation and frame of reference ===
Similar to the issue of a point of reference is the issue of orientation: a displacement in 2 or 3 dimensions is not just a length, but is a length together with a ''direction''. (In 1 dimension, this issue is equivalent to the distinction between positive and negative.) Thus, to compare or combine two dimensional quantities in multi-dimensional Euclidean space, one also needs a bearing: they need to be compared to a [[frame of reference]].

This leads to the [[#Extensions|extensions]] discussed below, namely Huntley's directed dimensions and Siano's orientational analysis.

=== Huntley's extensions ===
Huntley has pointed out that a dimensional analysis can become more powerful by discovering new independent dimensions in the quantities under consideration, thus increasing the rank <math>m</math> of the dimensional matrix.<ref>{{harv|Huntley|1967}}</ref>

He introduced two approaches:
* The magnitudes of the components of a vector are to be considered dimensionally independent. For example, rather than an undifferentiated length dimension L, we may have L<sub>x</sub> represent dimension in the x-direction, and so forth. This requirement stems ultimately from the requirement that each component of a physically meaningful equation (scalar, vector, or tensor) must be dimensionally consistent.
* Mass as a measure of the quantity of matter is to be considered dimensionally independent from mass as a measure of inertia.

==== Directed dimensions ====
As an example of the usefulness of the first approach, suppose we wish to calculate the [[trajectory#Range and height|distance a cannonball travels]] when fired with a vertical velocity component <math>v_\text{y}</math> and a horizontal velocity component {{tmath|v_\text{x} }}, assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then {{mvar|R}}, the distance travelled, with dimension L, {{tmath|v_\text{x} }}, {{tmath|v_\text{y} }}, both dimensioned as T<sup>−1</sup>L, and {{mvar|g}} the downward acceleration of gravity, with dimension T<sup>−2</sup>L.

With these four quantities, we may conclude that the equation for the range {{mvar|R}} may be written:
: <math>R \propto v_\text{x}^a\,v_\text{y}^b\,g^c .</math>

Or dimensionally
: <math>\mathsf{L} = \left(\mathsf{T}^{-1}\mathsf{L}\right)^{a+b} \left(\mathsf{T}^{-2}\mathsf{L}\right)^c</math>
from which we may deduce that <math>a + b + c = 1</math> and {{tmath|1=a + b + 2c = 0}}, which leaves one exponent undetermined. This is to be expected since we have two fundamental dimensions T and L, and four parameters, with one equation.

However, if we use directed length dimensions, then <math>v_\mathrm{x}</math> will be dimensioned as T<sup>−1</sup>L<sub>{{math|x}}</sub>, <math>v_\mathrm{y}</math> as T<sup>−1</sup>L<sub>{{math|y}}</sub>, {{mvar|R}} as L<sub>{{math|x}}</sub> and {{mvar|g}} as T<sup>−2</sup>L<sub>{{math|y}}</sub>. The dimensional equation becomes:
: <math>
  \mathsf{L}_\mathrm{x} =
    \left({\mathsf{T}^{-1}}{\mathsf{L}_\mathrm{x}}\right)^a
    \left({\mathsf{T}^{-1}}{\mathsf{L}_\mathrm{y}}\right)^b 
    \left({\mathsf{T}^{-2}}{\mathsf{L}_\mathrm{y}}\right)^c
</math>
and we may solve completely as {{math|1=''a'' = 1}}, {{math|1=''b'' = 1}} and {{math|1=''c'' = −1}}. The increase in deductive power gained by the use of directed length dimensions is apparent.

Huntley's concept of directed length dimensions however has some serious limitations:
* It does not deal well with vector equations involving the ''[[cross product]]'',
* nor does it handle well the use of ''angles'' as physical variables.

It also is often quite difficult to assign the L, L<sub>{{math|x}}</sub>, L<sub>{{math|y}}</sub>, L<sub>{{math|z}}</sub>, symbols to the physical variables involved in the problem of interest. He invokes a procedure that involves the "symmetry" of the physical problem. This is often very difficult to apply reliably: It is unclear as to what parts of the problem that the notion of "symmetry" is being invoked. Is it the symmetry of the physical body that forces are acting upon, or to the points, lines or areas at which forces are being applied? What if more than one body is involved with different symmetries?

Consider the spherical bubble attached to a cylindrical tube, where one wants the flow rate of air as a function of the pressure difference in the two parts. What are the Huntley extended dimensions of the viscosity of the air contained in the connected parts? What are the extended dimensions of the pressure of the two parts? Are they the same or different? These difficulties are responsible for the limited application of Huntley's directed length dimensions to real problems.

==== Quantity of matter ====
In Huntley's second approach, he holds that it is sometimes useful (e.g., in fluid mechanics and thermodynamics) to distinguish between mass as a measure of inertia (''inertial mass''), and mass as a measure of the quantity of matter. '''Quantity of matter''' is defined by Huntley as a quantity only {{em|proportional}} to inertial mass, while not implicating inertial properties. No further restrictions are added to its definition.

For example, consider the derivation of [[Poiseuille's Law]]. We wish to find the rate of mass flow of a viscous fluid through a circular pipe. Without drawing distinctions between inertial and substantial mass, we may choose as the relevant variables:
{| class="wikitable"
! Symbol !! Variable !! Dimension
|-
| <math>\dot{m}</math> || mass flow rate || T<sup>−1</sup>M
|-
| <math>p_\text{x}</math> || pressure gradient along the pipe || T<sup>−2</sup>L<sup>−2</sup>M
|-
| {{mvar|ρ}} || density || L<sup>−3</sup>M
|-
| {{mvar|η}} || dynamic fluid viscosity || T<sup>−1</sup>L<sup>−1</sup>M
|-
| {{mvar|r}} || radius of the pipe || L
|}

There are three fundamental variables, so the above five equations will yield two independent dimensionless variables:
: <math>\pi_1 = \frac{\dot{m}}{\eta r}</math>
: <math>\pi_2 = \frac{p_\mathrm{x}\rho r^5}{\dot{m}^2}</math>

If we distinguish between inertial mass with dimension <math>M_\text{i}</math> and quantity of matter with dimension <math>M_\text{m}</math>, then mass flow rate and density will use quantity of matter as the mass parameter, while the pressure gradient and coefficient of viscosity will use inertial mass. We now have four fundamental parameters, and one dimensionless constant, so that the dimensional equation may be written:
: <math>C = \frac{p_\mathrm{x}\rho r^4}{\eta \dot{m}}</math>

where now only {{mvar|C}} is an undetermined constant (found to be equal to <math>\pi/8</math> by methods outside of dimensional analysis). This equation may be solved for the mass flow rate to yield [[Poiseuille's law]].

Huntley's recognition of quantity of matter as an independent quantity dimension is evidently successful in the problems where it is applicable, but his definition of quantity of matter is open to interpretation, as it lacks specificity beyond the two requirements he postulated for it. For a given substance, the SI dimension [[amount of substance]], with unit [[Mole (unit)|mole]], does satisfy Huntley's two requirements as a measure of quantity of matter, and could be used as a quantity of matter in any problem of dimensional analysis where Huntley's concept is applicable.

=== 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.