Dimensional analysis: Difference between revisions

'''''Commensurable''''' physical quantities are of the same [[Kind of quantity|kind]] and have the same dimension, and can be directly compared to each other, even if they are expressed in differing units of measurement; e.g., metres and feet, grams and pounds, seconds and years. ''Incommensurable'' physical [[Quantity|quantities]] are of different [[Kind of quantity|kinds]] and have different dimensions, and can not be directly compared to each other, no matter what [[units]] they are expressed in, e.g. metres and grams, seconds and grams, metres and seconds. For example, asking whether a gram is larger than an hour is meaningless.

Any physically meaningful [[equation]], or [[inequality (mathematics)|inequality]], ''must'' have the same dimensions on its left and right sides, a property known as ''dimensional homogeneity''. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on [[Formal proof|derived]] equations and [[Computation|computations]]. It also serves as a guide and constraint in deriving equations that may describe a physical [[system]] in the absence of a more rigorous derivation.

The concept of '''physical dimension''', and of dimensional analysis, was introduced by [[Joseph Fourier]] in 1822.<ref name="Bolster">{{cite journal|lastlast1=Bolster|firstfirst1=Diogo|last2=Hershberger|first2=Robert E.|last3=Donnelly|first3=Russell E.|title=Dynamic similarity, the dimensionless science|url=https://pubs.aip.org/physicstoday/article-abstract/64/9/42/413713/Dynamic-similarity-the-dimensionless|workjournal=Physics Today|doi=10.1063/PT.3.1258|date=September 2011|volume=64|issue=9|pages=42-4742–47|url-access=subscription}}</ref>{{rp|42}}

A dimensional equation can have the dimensions reduced or eliminated through [[nondimensionalization]], which begins with dimensional analysis, and involves scaling quantities by [[characteristic units]] of a system or [[physical constantsconstant]]s of nature.<ref name="Bolster"/>{{rp|43}} This may give insight into the fundamental properties of the system, as illustrated in the examples below.

: <math>\operatorname{dim}v
= \frac{\text{length}}{\text{time}}
= \frac{\mathsf{L}}{\mathsf{T}}
= \mathsf{T}^{-1}\mathsf{L} .</math>

: <math>\operatorname{dim}F
= \text{mass} \times \text{acceleration}
= \textmathsf{massM} \times \fracmathsf{\text{length}T}^{\text{time}^-2} = \frac{\mathsf{L}\mathsf{M}}{\mathsf{T}^2}
= \mathsf{T}^{-2}\mathsf{L}\mathsf{M} .</math>

: <math>\begin{align}\operatorname{dim}P

&= \frac{\text{force}}{\text{area}}\\

&= \frac{\textmathsf{massT}^{-2} \times mathsf{L}\textmathsf{accelerationM}}{\textmathsf{areaL}^2}\\

&= \frac{\mathsf{M L T}^{-2}}}{\mathsf{L}^2{-1}\mathsf{M}\\ .</math>

: <math>\begin{align}\operatorname{dim}E

&= \text{force} \times \text{displacement}\\

&= \textmathsf{massT}^{-2} \times mathsf{L}\textmathsf{accelerationM} \times \textmathsf{displacementL}\\

: <math>\begin{align}\operatorname{dim}P

&= \frac{\text{energy}}{\text{time}}\\

&= \frac{\mathsf{T}^{-2}}\mathsf{L}^2}\mathsf{M}}{\mathsf{T}}\\

&= \mathsf{T}^{-3}}\mathsf{L}^2}\mathsf{M} .\\</math>

: <math>\begin{align}\operatorname{dim}Q

= \endmathsf{alignT}\mathsf{I} .</math>

The dimension of the physical quantity [[electric potential differencevoltage]] {{math|''V''}} is

: <math>\begin{align}\operatorname{dim}V

&= \frac{\mathsftext{T^{-3power}}\mathsf{L^2}\mathsftext{Mcurrent} }{\mathsf{I}}\\

&= \frac{\mathsf{T}^{-3}}\mathsf{L}^2}\mathsf{M} }{\mathsf{I^{-1}}.\\

: <math>\begin{alignat}{2}\operatorname{dim}C

&= \frac{\text{electric charge}}{\text{electric potential difference}}\\

&= \frac {\mathsf{T}\mathsf{I}}{\mathsf{T}^{-3}}\mathsf{L}^2}\mathsf{M} \mathsf{I}^{-1}}}\\

&= \frac{\mathsf{T^4}\mathsf{IL^{-2}}{\mathsf{LM^2{-1}}\mathsf{MI^2}}\\ .</math>

# If {{math|''R''}} is a variable that depends upon independent variables {{math|''R''₁}}, {{math|''R''₂}}, {{math|''R''₃}}, ..., {{math|''R''_''n''}}, then the [[functional equation]] can be written as {{math||1=''R'' = ''F''(''R''₁, ''R''₂, ''R''₃, ..., ''R''_''n'')}}.

A set of [[Base unit (measurement)|base unit]]s for a [[system of measurement]] is a conventionally chosen set of units, none of which can be expressed as a combination of the others and in terms of which all the remaining units of the system can be expressed.<ref>{{Cite bookcitation |lastauthor=JCGM 200|author-link=Joint Committee for Guides in Metrology |url=https://www.bipm.org/utils/common/documents/jcgm/JCGM_200_2012.pdf |title=JCGM 200:2012 – International vocabulary of metrology – Basic and general concepts and associated terms (VIM) |year=2012 |edition=3rd |access-date=2 June 2015 |archive-url=https://web.archive.org/web/20150923224356/http://www.bipm.org/utils/common/documents/jcgm/JCGM_200_2012.pdf |archive-date=23 September 2015 |url-status=dead}}</ref> For example, units for [[length]] and time are normally chosen as base units. Units for [[volume]], however, can be factored into the base units of length (m³), thus they are considered derived or compound units.

* the integral of force with respect to the distance ({{math|''s''}}) the object has travelled (<math>{{tmath|\textstyle\int F\ ds</math>}}, [[Work (physics)#Mathematical calculation|work]]) has dimension {{dimanalysis|mass=1|length=2|time=−2}}.

A related principle is that any physical law that accurately describes the real world must be independent of the units used to measure the physical variables.<ref>{{Cite book |last1=de Jong |first1=Frits J. |url=https://archive.org/details/dimensionalanaly0000jong |title=Dimensional analysis for economists |last2=Quade |first2=Wilhelm |publisher=North Holland |year=1967 |page=[https://archive.org/details/dimensionalanaly0000jong/page/28 28] |url-access=registration}}</ref> For example, [[Newton's laws of motion]] must hold true whether distance is measured in miles or kilometres. This principle gives rise to the form that a conversion factor between atwo unitunits that measuresmeasure the same dimension must take: multiplication by a simple constant. It also ensures equivalence; for example, if two buildings are the same height in feet, then they must be the same height in metres.

A simple application of dimensional analysis to mathematics is in computing the form of the [[N-sphere#Volume of the n-ball|volume of an {{math|''n''}}-ball]] (the solid ball in ''n'' dimensions), or the area of its surface, the [[n-sphere|{{math|''n''}}-sphere]]: being an {{math|''n''}}-dimensional figure, the volume scales as {{math|''x''{{sup|''n''}}}}, while the surface area, being {{math|(''n'' − 1)}}-dimensional, scales as {{math|''x''{{sup|''n''−1}}}}. Thus the volume of the {{math|''n''}}-ball in terms of the radius is {{math|''C''{{isupsub|''n''}}''r''{{isup|''n''}}}}, for some constant {{math|''C''{{isupsub|''n''}}}}. Determining the constant takes more involved mathematics, but the form can be deduced and checked by dimensional analysis alone.

[[James Clerk Maxwell]] played a major role in establishing modern use of dimensional analysis by distinguishing mass, length, and time as fundamental units, while referring to other units as derived.<ref name="maxwell">{{Citation |last=Roche |first=John J |title=The Mathematics of Measurement: A Critical History |page=203 |year=1998 |publisher=Springer |isbn=978-0-387-91581-4 |url= https://books.google.com/books?id=eiQOqS-Q6EkC&pg=PA203|quote = Beginning apparently with Maxwell, mass, length and time began to be interpreted as having a privileged fundamental character and all other quantities as derivative, not merely with respect to measurement, but with respect to their physical status as well.}}</ref> Although Maxwell defined length, time and mass to be "the three fundamental units", he also noted that gravitational mass can be derived from length and time by assuming a form of [[Newton's law of universal gravitation]] in which the [[gravitational constant]] {{math|''G''}} is taken as [[1|unity]], thereby defining {{nowrap|1=M = T⁻²L³}}.<ref name="maxwell2">{{Citation |last=Maxwell |first=James Clerk |title=A Treatise on Electricity and Magnetism |page=4 |year=1873}}</ref> By assuming a form of [[Coulomb's law]] in which the [[Coulomb constant]] ''k''_e is taken as unity, Maxwell then determined that the dimensions of an electrostatic unit of charge were {{nowrap|1=Q = T⁻¹L^3/2M^1/2}},<ref name="maxwell3">{{Citation |last= Maxwell |first=James Clerk |title=A Treatise on Electricity and Magnetism |series=Clarendon Press series |page=45 |year=1873 |publisher=Oxford |hdl=2027/uc1.l0065867749 |hdl-access=free}}</ref> which, after substituting his {{nowrap|1=M = T⁻²L³}} equation for mass, results in charge having the same dimensions as mass, viz. {{nowrap|1=Q = T⁻²L³}}.

The variable {{mvar|g}} does not occur in the group. It is easy to see that it is impossible to form a dimensionless product of powers that combines {{mvar|g}} with {{mvar|k}}, {{mvar|m}}, and {{mvar|T}}, because {{mvar|g}} is the only quantity that involves the dimension L. This implies that in this problem the {{math|''g''}} is irrelevant. Dimensional analysis can sometimes yield strong statements about the ''irrelevance'' of some quantities in a problem, or the need for additional parameters. If we have chosen enough variables to properly describe the problem, then from this argument we can conclude that the period of the mass on the spring is independent of {{math|''g''}}: it is the same on the earth or the moon. The equation demonstrating the existence of a product of powers for our problem can be written in an entirely equivalent way: <math>{{tmath|1=T = \kappa \sqrt\tfrac{m}{k}</math> }}, for some dimensionless constant {{math|''κ''}} (equal to <math>\sqrt{C}</math> from the original dimensionless equation).

{{Mainfurther|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).

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''. (This issue does not arise inIn 1 dimension, orthis ratherissue is equivalent to the distinction between positive and negative.) Thus, to compare or combine two dimensional quantities in a multi-dimensional Euclidean space, one also needs ana orientationbearing: they need to be compared to a [[frame of reference]].

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_v_\mathrmtext{y}</math> and a horizontal velocity component <math>V_{{tmath|v_\mathrmtext{x}</math> }}, 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, <math>V_{{tmath|v_\mathrmtext{x}</math> }}, <math>V_{{tmath|v_\mathrmtext{y}</math> }}, both dimensioned as T⁻¹L, and {{mvar|g}} the downward acceleration of gravity, with dimension T⁻²L.

: <math>R \propto V_v_\text{x}^a\,V_v_\text{y}^b\,g^c .\,</math>

: <math>\mathsf{L} = \left(\frac{\mathsf{LT}^{-1}{\mathsf{T}L}\right)^{a+b} \left(\frac{\mathsf{LT}^{-2}{\mathsf{T}^2L}\right)^c\,</math>

from which we may deduce that <math>a + b + c = 1</math> and <math>{{tmath|1=a + b + 2c = 0</math>}}, 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_v_\mathrm{x}</math> will be dimensioned as T⁻¹L_{{math|x}}, <math>V_v_\mathrm{y}</math> as T⁻¹L_{{math|y}}, {{mvar|R}} as L_{{math|x}} and {{mvar|g}} as T⁻²L_{{math|y}}. The dimensional equation becomes:

\left(\frac{\mathsf{LT}_\mathrm^{x-1}}{\mathsf{TL}_\mathrm{x}}\right)^a

\left(\frac{\mathsf{LT}_\mathrm^{y-1}}{\mathsf{TL}_\mathrm{y}}\right)^b

\left(\frac{\mathsf{LT}_\mathrm^{y-2}}{\mathsf{TL}_\mathrm{y}^2}\right)^c

and we may solve completely as <{{math>|1=''a'' = 1</math>}}, <{{math>|1=''b'' = 1</math>}} and <{{math>|1=''c'' = -1</math>−1}}. The increase in deductive power gained by the use of directed length dimensions is apparent.

[[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 }}</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''²}}, but offers no insight into the relationship between {{math|''R''}} and {{math|''θ''}}.

which for <math>a = \theta</math> and <math>b = \pi/2</math> yields <math>{{tmath|1=\sin(\theta\,1_\text{z} + [\pi/2]\,1_\text{z}) = 1_\text{z}\cos(\theta\,1_\text{z})</math>}}. 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 <math>{{tmath|1_0</math>}}.

Dimensional homogeneity will now correctly yield {{math|1=''a'' = −1}} and {{math|1=''b'' = 2}}, and orientational homogeneity requires that <math>{{tmath|1=1_x /(1_y^a 1_z^c)=1_z^{c+1} = 1</math>}}. 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}}.

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| chapter-url=https://books.google.com/books?id=NeWVeylbeiQC&pg=PA186

| editor-last=Lewis | editor-first=John G.

| title=Proceedings of the Fifth SIAM Conference on Applied Linear Algebra

| publisher=SIAM
| isbn=978-0-89871-336-7
| pages=186–190
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