Editing Dimensional analysis (section)

== Formulation ==
{{redirect|Dimension (physics)|physical dimensions|Size}}

The [[Buckingham π theorem]] describes how every physically meaningful equation involving {{math|''n''}} variables can be equivalently rewritten as an equation of {{math|''n'' − ''m''}} dimensionless parameters, where ''m'' is the [[rank of a matrix|rank]] of the dimensional [[matrix (mathematics)|matrix]]. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.

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 constant]]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.

The dimension of a [[physical quantity]] can be expressed as a product of the base physical dimensions such as length, mass and time, each raised to an integer (and occasionally [[rational number|rational]]) [[power (mathematics)|power]]. The ''dimension'' of a physical quantity is more fundamental than some ''scale'' or [[units of measurement|unit]] used to express the amount of that physical quantity. For example, ''mass'' is a dimension, while the kilogram is a particular reference quantity chosen to express a quantity of mass. The choice of unit is arbitrary, and its choice is often based on historical precedent. [[Natural units]], being based on only universal constants, may be thought of as being "less arbitrary".

There are many possible choices of base physical dimensions. The [[SI standard]] selects the following dimensions and corresponding '''dimension symbols''':{{anchor|Dimension symbol}} 
: [[time]] (T), [[length]] (L), [[mass]] (M), [[electric current]] (I), [[absolute temperature]] (Θ), [[amount of substance]] (N) and [[luminous intensity]] (J).
The symbols are by convention usually written in [[roman type|roman]] [[sans serif]] typeface.<ref name=SIBrochure9th>{{cite book |author1=BIPM |author1-link=BIPM |title=SI Brochure: The International System of Units (SI) |date=2019 |isbn=978-92-822-2272-0 |pages=136–137 |edition=v. 1.08, 9th |url=https://www.bipm.org/en/publications/si-brochure |access-date=1 September 2021 |language=en, fr |format=PDF |chapter=2.3.3 Dimensions of quantities}}</ref> Mathematically, the dimension of the quantity {{math|''Q''}} is given by 
: <math>\operatorname{dim}Q = \mathsf{T}^a\mathsf{L}^b\mathsf{M}^c\mathsf{I}^d\mathsf{\Theta}^e\mathsf{N}^f\mathsf{J}^g</math>
where {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, {{math|''d''}}, {{math|''e''}}, {{math|''f''}}, {{math|''g''}} are the dimensional exponents. Other physical quantities could be defined as the base quantities, as long as they form a [[basis (linear algebra)|basis]] – for instance, one could replace the dimension (I) of [[electric current]] of the SI basis with a dimension (Q) of [[electric charge]], since {{nowrap|1=Q = TI}}.

A quantity that has only {{math|''b'' ≠ 0}} (with all other exponents zero) is known as a '''[[geometric]] quantity'''.  A quantity that has only both {{math|''a'' ≠ 0}} and {{math|''b'' ≠ 0}} is known as a '''[[kinematic]] quantity'''. A quantity that has only all of {{math|''a'' ≠ 0}}, {{math|''b'' ≠ 0}}, and {{math|''c'' ≠ 0}} is known as a '''[[dynamics (mechanics)|dynamic]] quantity'''.<ref>{{cite book | chapter-url=https://link.springer.com/chapter/10.1007/978-1-349-00245-0_1 | doi=10.1007/978-1-349-00245-0_1 | chapter=Principles of the Theory of Dimensions | title=Theory of Hydraulic Models | year=1971 | last1=Yalin | first1=M. Selim | pages=1–34 | isbn=978-1-349-00247-4 }}</ref>
A quantity that has all exponents null is said to have '''dimension one'''.<ref name=SIBrochure9th/>

The unit chosen to express a physical quantity and its dimension are related, but not identical concepts. The units of a physical quantity are defined by convention and related to some standard; e.g., length may have units of metres, feet, inches, miles or micrometres; but any length always has a dimension of L, no matter what units of length are chosen to express it. Two different units of the same physical quantity have [[conversion factors]] that relate them. For example, {{nowrap|1=1 in = 2.54 cm}}; in this case 2.54&nbsp;cm/in is the conversion factor, which is itself dimensionless. Therefore, multiplying by that conversion factor does not change the dimensions of a physical quantity.

There are also physicists who have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity,<ref name="duff">{{Citation |last1=Duff |first1=M.J. |last2=Okun |first2=L.B. |last3=Veneziano |first3=G. |title=Trialogue on the number of fundamental constants |journal=Journal of High Energy Physics |volume=2002 |page=023 |date=September 2002 |doi=10.1088/1126-6708/2002/03/023 |arxiv=physics/0110060|bibcode = 2002JHEP...03..023D |issue=3 |s2cid=15806354 }}</ref> although this does not invalidate the usefulness of dimensional analysis.

=== Simple cases ===
As examples, the dimension of the physical quantity [[speed]] {{math|''v''}} is
: <math>\operatorname{dim}v
= \frac{\text{length}}{\text{time}}
= \frac{\mathsf{L}}{\mathsf{T}}
= \mathsf{T}^{-1}\mathsf{L} .</math>

The dimension of the physical quantity [[acceleration]] {{math|''a''}} is
: <math>\operatorname{dim}a
= \frac{\text{speed}}{\text{time}}
= \frac{\mathsf{T}^{-1}\mathsf{L}}{\mathsf{T}}
= \mathsf{T}^{-2}\mathsf{L} .</math>

The dimension of the physical quantity [[force]] {{math|''F''}} is
: <math>\operatorname{dim}F
= \text{mass} \times \text{acceleration}
= \mathsf{M} \times \mathsf{T}^{-2}\mathsf{L}
= \mathsf{T}^{-2}\mathsf{L}\mathsf{M} .</math>

The dimension of the physical quantity [[pressure]] {{math|''P''}} is 
: <math>\operatorname{dim}P
= \frac{\text{force}}{\text{area}}
= \frac{\mathsf{T}^{-2}\mathsf{L}\mathsf{M}}{\mathsf{L}^2}
= \mathsf{T}^{-2}\mathsf{L}^{-1}\mathsf{M} .</math>

The dimension of the physical quantity [[energy]] {{math|''E''}} is
: <math>\operatorname{dim}E
= \text{force} \times \text{displacement}
= \mathsf{T}^{-2}\mathsf{L}\mathsf{M} \times \mathsf{L}
= \mathsf{T}^{-2}\mathsf{L}^2\mathsf{M} .</math>

The dimension of the physical quantity [[Power (physics)|power]] {{math|''P''}} is
: <math>\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>

The dimension of the physical quantity [[electric charge]] {{math|''Q''}} is
: <math>\operatorname{dim}Q
= \text{current} \times \text{time}
= \mathsf{T}\mathsf{I} .</math>

The dimension of the physical quantity [[voltage]] {{math|''V''}} is
: <math>\operatorname{dim}V
= \frac{\text{power}}{\text{current}}
= \frac{\mathsf{T}^{-3}\mathsf{L}^2\mathsf{M}}{\mathsf{I}}
= \mathsf{T^{-3}}\mathsf{L}^2\mathsf{M} \mathsf{I}^{-1} .</math>

The dimension of the physical quantity [[capacitance]] {{math|''C''}} is
: <math>\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}}
= \mathsf{T^4}\mathsf{L^{-2}}\mathsf{M^{-1}}\mathsf{I^2} .</math>

=== Rayleigh's method ===
In dimensional analysis, '''Rayleigh's method''' is a conceptual tool used in [[physics]], [[chemistry]], and [[engineering]]. It expresses a [[functional relationship]] of some [[variable (mathematics)|variables]] in the form of an [[exponential equation]]. It was named after [[Lord Rayleigh]].

The method involves the following steps:
# Gather all the [[independent variable]]s that are likely to influence the [[dependent variable]].
# If {{math|''R''}} is a variable that depends upon independent variables {{math|''R''<sub>1</sub>}}, {{math|''R''<sub>2</sub>}}, {{math|''R''<sub>3</sub>}}, ..., {{math|''R''<sub>''n''</sub>}}, then the [[functional equation]] can be written as {{math|1=''R'' = ''F''(''R''<sub>1</sub>, ''R''<sub>2</sub>, ''R''<sub>3</sub>, ..., ''R''<sub>''n''</sub>)}}.
# Write the above equation in the form {{math|1=''R'' = ''C'' ''R''<sub>1</sub><sup>''a''</sup> ''R''<sub>2</sub><sup>''b''</sup> ''R''<sub>3</sub><sup>''c''</sup> ... ''R''<sub>''n''</sub><sup>''m''</sup>}}, where {{math|''C''}} is a [[dimensionless constant]] and {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, ..., {{math|''m''}} are arbitrary exponents.
# Express each of the quantities in the equation in some [[Base unit (measurement)|base unit]]s in which the solution is required.
# By using [[#Dimensional homogeneity|dimensional homogeneity]], obtain a [[set (mathematics)|set]] of [[simultaneous equations]] involving the exponents {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, ..., {{math|''m''}}.
# [[Equation solving|Solve]] these equations to obtain the values of the exponents {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, ..., {{math|''m''}}.
# [[Simultaneous equations#Substitution method|Substitute]] the values of exponents in the main equation, and form the [[non-dimensional]] [[parameter]]s by [[Combining like terms|grouping]] the variables with like exponents.

As a drawback, Rayleigh's method does not provide any information regarding number of dimensionless groups to be obtained as a result of dimensional analysis.