
In
engineering
Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
,
applied mathematics
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...
, and
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, the Buckingham theorem is a key
theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
in
dimensional analysis. It is a formalisation of
Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically meaningful equation involving a certain number ''n'' physical variables, then the original equation can be rewritten in terms of a set of ''p'' = ''n'' − ''k'' dimensionless parameters
1,
2, ...,
''p'' constructed from the original variables, where ''k'' is the number of physical dimensions involved; it is obtained as the
rank of a particular
matrix.
The theorem provides a method for computing sets of dimensionless parameters from the given variables, or
nondimensionalization, even if the form of the equation is still unknown.
The Buckingham theorem indicates that validity of the
laws of physics does not depend on a specific
unit system. A statement of this theorem is that any physical law can be expressed as an
identity involving only dimensionless combinations (ratios or products) of the variables linked by the law (for example, pressure and volume are linked by
Boyle's law
Boyle's law, also referred to as the Boyle–Mariotte law or Mariotte's law (especially in France), is an empirical gas laws, gas law that describes the relationship between pressure and volume of a confined gas. Boyle's law has been stated as:
...
– they are
inversely proportional). If the dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and the theorem would not hold.
History
Although named for
Edgar Buckingham, the theorem was first proved by the French mathematician
Joseph Bertrand in 1878. Bertrand considered only special cases of problems from electrodynamics and heat conduction, but his article contains, in distinct terms, all the basic ideas of the modern proof of the theorem and clearly indicates the theorem's utility for modelling physical phenomena. The technique of using the theorem ("the method of dimensions") became widely known due to the works of
Rayleigh. The first application of the theorem ''in the general case''
[When in applying the –theorem there arises an ''arbitrary function'' of dimensionless numbers.] to the dependence of pressure drop in a pipe upon governing parameters probably dates back to 1892, a heuristic proof with the use of series expansions, to 1894.
Formal generalization of the theorem for the case of arbitrarily many quantities was given first by in 1892, then in 1911—apparently independently—by both A. Federman and
D. Riabouchinsky, and again in 1914 by Buckingham. It was Buckingham's article that introduced the use of the symbol "
" for the dimensionless variables (or parameters), and this is the source of the theorem's name.
Statement
More formally, the number
of dimensionless terms that can be formed is equal to the
nullity of the
dimensional matrix, and
is the
rank. For experimental purposes, different systems that share the same description in terms of these
dimensionless numbers are equivalent.
In mathematical terms, if we have a physically meaningful equation such as
where
are any
physical variables, and there is a maximal dimensionally independent subset of size
,
[A dimensionally independent set of variables is one for which the only exponents yielding a dimensionless quantity are . This is precisely the notion of ]linear independence
In the theory of vector spaces, a set (mathematics), set of vector (mathematics), vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then th ...
. then the above equation can be restated as
where
are dimensionless parameters constructed from the
by
dimensionless equations — the so-called ''Pi groups'' — of the form
where the exponents
are rational numbers. (They can always be taken to be integers by redefining
as being raised to a power that clears all denominators.) If there are
fundamental units in play, then
.
Significance
The Buckingham theorem provides a method for computing sets of dimensionless parameters from given variables, even if the form of the equation remains unknown. However, the choice of dimensionless parameters is not unique; Buckingham's theorem only provides a way of generating sets of dimensionless parameters and does not indicate the most "physically meaningful".
Two systems for which these parameters coincide are called ''similar'' (as with
similar triangles, they differ only in scale); they are equivalent for the purposes of the equation, and the experimentalist who wants to determine the form of the equation can choose the most convenient one. Most importantly, Buckingham's theorem describes the relation between the number of variables and fundamental dimensions.
Proof
For simplicity, it will be assumed that the space of fundamental and derived physical units forms a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over the
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, with the fundamental units as basis vectors, and with multiplication of physical units as the "vector addition" operation, and raising to powers as the "scalar multiplication" operation:
represent a dimensional variable as the set of exponents needed for the fundamental units (with a power of zero if the particular fundamental unit is not present). For instance, the
standard gravity has units of
(length over time squared), so it is represented as the vector
with respect to the basis of fundamental units (length, time). We could also require that exponents of the fundamental units be rational numbers and modify the proof accordingly, in which case the exponents in the pi groups can always be taken as rational numbers or even integers.
Rescaling units
Suppose we have quantities
, where the units of
contain length raised to the power
. If we originally measure length in meters but later switch to centimeters, then the numerical value of
would be rescaled by a factor of
. Any physically meaningful law should be invariant under an arbitrary rescaling of every fundamental unit; this is the fact that the pi theorem hinges on.
Formal proof
Given a system of
dimensional variables
in
fundamental (basis) dimensions, the ''dimensional matrix'' is the
matrix
whose
rows correspond to the fundamental dimensions and whose
columns are the dimensions of the variables: the
th entry (where
and
) is the power of the
th fundamental dimension in the
th variable.
The matrix can be interpreted as taking in a combination of the variable quantities and giving out the dimensions of the combination in terms of the fundamental dimensions. So the
(column) vector that results from the multiplication
consists of the units of
in terms of the
fundamental independent (basis) units.
If we rescale the
th fundamental unit by a factor of
, then
gets rescaled by
, where
is the
th entry of the dimensional matrix. In order to convert this into a linear algebra problem, we take
logarithms (the base is irrelevant), yielding
which is an
action of
on
. We define a physical law to be an arbitrary function
such that
is a permissible set of values for the physical system when
. We further require
to be invariant under this action. Hence it descends to a function
. All that remains is to exhibit an isomorphism between
and
, the (log) space of pi groups
.
We construct an
matrix
whose columns are a basis for
. It tells us how to embed
into
as the kernel of
. That is, we have an
exact sequence
:
Taking tranposes yields another exact sequence
:
The
first isomorphism theorem produces the desired isomorphism, which sends the
coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
to
. This corresponds to rewriting the tuple
into the pi groups
coming from the columns of
.
The
International System of Units
The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official s ...
defines seven base units, which are the
ampere
The ampere ( , ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to 1 c ...
,
kelvin,
second
The second (symbol: s) is a unit of time derived from the division of the day first into 24 hours, then to 60 minutes, and finally to 60 seconds each (24 × 60 × 60 = 86400). The current and formal definition in the International System of U ...
,
metre
The metre (or meter in US spelling; symbol: m) is the base unit of length in the International System of Units (SI). Since 2019, the metre has been defined as the length of the path travelled by light in vacuum during a time interval of of ...
,
kilogram
The kilogram (also spelled kilogramme) is the base unit of mass in the International System of Units (SI), equal to one thousand grams. It has the unit symbol kg. The word "kilogram" is formed from the combination of the metric prefix kilo- (m ...
,
candela
The candela (symbol: cd) is the unit of luminous intensity in the International System of Units (SI). It measures luminous power per unit solid angle emitted by a light source in a particular direction. Luminous intensity is analogous to radi ...
and
mole. It is sometimes advantageous to introduce additional base units and techniques to refine the technique of dimensional analysis. (See
orientational analysis and reference.
)
Examples
Speed
This example is elementary but serves to demonstrate the procedure.
Suppose a car is driving at 100 km/h; how long does it take to go 200 km?
This question considers
dimensioned variables: distance
time
and speed
and we are seeking some law of the form
Any two of these variables are dimensionally independent, but the three taken together are not. Thus there is
dimensionless quantity.
The dimensional matrix is
in which the rows correspond to the basis dimensions
and
and the columns to the considered dimensions
where the latter stands for the speed dimension. The elements of the matrix correspond to the powers to which the respective dimensions are to be raised. For instance, the third column
states that
represented by the column vector
is expressible in terms of the basis dimensions as
since
For a dimensionless constant
we are looking for vectors