Inengineering andscience,dimensional analysis is the analysis of the relationships between differentphysical quantities by identifying theirbase quantities (such aslength,mass,time, andelectric current) andunits of measurement (such as metres and grams) and tracking these dimensions as calculations or comparisons are performed. The term dimensional analysis is also used to refer toconversion of units from one dimensional unit to another, which can be used to evaluate scientific formulae.
Commensurable physical quantities are of the samekind 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 physicalquantities are of differentkinds and have different dimensions, and can not be directly compared to each other, no matter whatunits 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 meaningfulequation, orinequality,must have the same dimensions on its left and right sides, a property known asdimensional homogeneity. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check onderived equations andcomputations. It also serves as a guide and constraint in deriving equations that may describe a physicalsystem in the absence of a more rigorous derivation.
The concept ofphysical dimension orquantity dimension, and of dimensional analysis, was introduced byJoseph Fourier in 1822.[1]: 42
TheBuckingham π theorem describes how every physically meaningful equation involvingn variables can be equivalently rewritten as an equation ofn −m dimensionless parameters, wherem is therank of the dimensionalmatrix. 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 throughnondimensionalization, which begins with dimensional analysis, and involves scaling quantities bycharacteristic units of a system orphysical constants of nature.[1]: 43 This may give insight into the fundamental properties of the system, as illustrated in the examples below.
The dimension of aphysical quantity can be expressed as a product of the base physical dimensions such as length, mass and time, each raised to an integer (and occasionallyrational)power. Thedimension of a physical quantity is more fundamental than somescale orunit 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. TheSI standard selects the following dimensions and correspondingdimension symbols:
The symbols are by convention usually written inromansans serif typeface.[2] Mathematically, the dimension of the quantityQ is given by
wherea,b,c,d,e,f,g are the dimensional exponents. Other physical quantities could be defined as the base quantities, as long as they form abasis – for instance, one could replace the dimension (I) ofelectric current of the SI basis with a dimension (Q) ofelectric charge, sinceQ = TI.
A quantity that has onlyb ≠ 0 (with all other exponents zero) is known as ageometric quantity. A quantity that has only botha ≠ 0 andb ≠ 0 is known as akinematic quantity. A quantity that has only all ofa ≠ 0,b ≠ 0, andc ≠ 0 is known as adynamic quantity.[3]A quantity that has all exponents null is said to havedimension one.[2]
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 haveconversion factors that relate them. For example,1 in = 2.54 cm; in this case 2.54 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,[4] although this does not invalidate the usefulness of dimensional analysis.
As examples, the dimension of the physical quantityspeedv is
The dimension of the physical quantityaccelerationa is
The dimension of the physical quantityforceF is
The dimension of the physical quantitypressureP is
The dimension of the physical quantityenergyE is
The dimension of the physical quantitypowerP is
The dimension of the physical quantityelectric chargeQ is
The dimension of the physical quantityvoltageV is
The dimension of the physical quantitycapacitanceC is
In dimensional analysis,Rayleigh's method is a conceptual tool used inphysics,chemistry, andengineering. It expresses afunctional relationship of somevariables in the form of anexponential equation. It was named afterLord Rayleigh.
The method involves the following steps:
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.
Many parameters and measurements in the physical sciences and engineering are expressed as aconcrete number—a numerical quantity and a corresponding dimensional unit. Often a quantity is expressed in terms of several other quantities; for example, speed is a combination of length and time, e.g. 60 kilometres per hour or 1.4 kilometres per second. Compound relations with "per" are expressed withdivision, e.g. 60 km/h. Other relations can involvemultiplication (often shown with acentered dot orjuxtaposition), powers (like m2 for square metres), or combinations thereof.
A set ofbase units for asystem 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.[5] For example, units forlength and time are normally chosen as base units. Units forvolume, however, can be factored into the base units of length (m3), thus they are considered derived or compound units.
Sometimes the names of units obscure the fact that they are derived units. For example, anewton (N) is a unit offorce, which may be expressed as the product of mass (with unit kg) and acceleration (with unit m⋅s−2). The newton is defined as1 N = 1 kg⋅m⋅s−2.
Percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. In other words, the % sign can be read as "hundredths", since1% = 1/100.
Taking a derivative with respect to a quantity divides the dimension by the dimension of the variable that is differentiated with respect to. Thus:
Likewise, taking an integral adds the dimension of the variable one is integrating with respect to, but in the numerator.
In economics, one distinguishes betweenstocks and flows: a stock has a unit (say, widgets or dollars), while a flow is a derivative of a stock, and has a unit of the form of this unit divided by one of time (say, dollars/year).
In some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions. For example,debt-to-GDP ratios are generally expressed as percentages: total debt outstanding (dimension of currency) divided by annual GDP (dimension of currency)—but one may argue that, in comparing a stock to a flow, annual GDP should have dimensions of currency/time (dollars/year, for instance) and thus debt-to-GDP should have the unit year, which indicates that debt-to-GDP is the number of years needed for a constant GDP to pay the debt, if all GDP is spent on the debt and the debt is otherwise unchanged.
The most basic rule of dimensional analysis is that of dimensional homogeneity.[6]
However, the dimensions form anabelian group under multiplication, so:
For example, it makes no sense to ask whether 1 hour is more, the same, or less than 1 kilometre, as these have different dimensions, nor to add 1 hour to 1 kilometre. However, it makes sense to ask whether 1 mile is more, the same, or less than 1 kilometre, being the same dimension of physical quantity even though the units are different. On the other hand, if an object travels 100 km in 2 hours, one may divide these and conclude that the object's average speed was 50 km/h.
The rule implies that in a physically meaningfulexpression only quantities of the same dimension can be added, subtracted, or compared. For example, ifmman,mrat andLman denote, respectively, the mass of some man, the mass of a rat and the length of that man, the dimensionally homogeneous expressionmman +mrat is meaningful, but the heterogeneous expressionmman +Lman is meaningless. However,mman/L2man is fine. Thus, dimensional analysis may be used as asanity check of physical equations: the two sides of any equation must be commensurable or have the same dimensions.
Even when two physical quantities have identical dimensions, it may nevertheless be meaningless to compare or add them. For example, althoughtorque and energy share the dimensionT−2L2M, they are fundamentally different physical quantities.
To compare, add, or subtract quantities with the same dimensions but expressed in different units, the standard procedure is first to convert them all to the same unit. For example, to compare 32 metres with 35 yards, use1 yard = 0.9144 m to convert 35 yards to 32.004 m.
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.[7] 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 two units that measure 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.
In dimensional analysis, a ratio which converts one unit of measure into another without changing the quantity is called aconversion factor. For example, kPa and bar are both units of pressure, and100 kPa = 1 bar. The rules of algebra allow both sides of an equation to be divided by the same expression, so this is equivalent to100 kPa / 1 bar = 1. Since any quantity can be multiplied by 1 without changing it, the expression "100 kPa / 1 bar" can be used to convert from bars to kPa by multiplying it with the quantity to be converted, including the unit. For example,5 bar × 100 kPa / 1 bar = 500 kPa because5 × 100 / 1 = 500, and bar/bar cancels out, so5 bar = 500 kPa.
Dimensional analysis is most often used in physics and chemistry – and in the mathematics thereof – but finds some applications outside of those fields as well.
A simple application of dimensional analysis to mathematics is in computing the form of thevolume of ann-ball (the solid ball inn dimensions), or the area of its surface, then-sphere: being ann-dimensional figure, the volume scales asxn, while the surface area, being(n − 1)-dimensional, scales asxn−1. Thus the volume of then-ball in terms of the radius isCnrn, for some constantCn. Determining the constant takes more involved mathematics, but the form can be deduced and checked by dimensional analysis alone.
In finance, economics, and accounting, dimensional analysis is most commonly referred to in terms of thedistinction between stocks and flows. More generally, dimensional analysis is used in interpreting variousfinancial ratios, economics ratios, and accounting ratios.
Influid mechanics, dimensional analysis is performed to obtain dimensionlesspi terms or groups. According to the principles of dimensional analysis, any prototype can be described by a series of these terms or groups that describe the behaviour of the system. Using suitable pi terms or groups, it is possible to develop a similar set of pi terms for a model that has the same dimensional relationships.[8] In other words, pi terms provide a shortcut to developing a model representing a certain prototype. Common dimensionless groups in fluid mechanics include:
The origins of dimensional analysis have been disputed by historians.[9][10] The first written application of dimensional analysis has been credited toFrançois Daviet, a student ofJoseph-Louis Lagrange, in a 1799 article at theTurin Academy of Science.[10]
This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually later formalized in theBuckingham π theorem.Simeon Poisson also treated the same problem of theparallelogram law by Daviet, in his treatise of 1811 and 1833 (vol I, p. 39).[11] In the second edition of 1833, Poisson explicitly introduces the termdimension instead of the Daviethomogeneity.
In 1822, the important Napoleonic scientistJoseph Fourier made the first credited important contributions[12] based on the idea that physical laws likeF =ma should be independent of the units employed to measure the physical variables.
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.[13] 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 ofNewton's law of universal gravitation in which thegravitational constantG is taken asunity, thereby definingM = T−2L3.[14] By assuming a form ofCoulomb's law in which theCoulomb constantke is taken as unity, Maxwell then determined that the dimensions of an electrostatic unit of charge wereQ = T−1L3/2M1/2,[15] which, after substituting hisM = T−2L3 equation for mass, results in charge having the same dimensions as mass, viz.Q = T−2L3.
Dimensional analysis is also used to derive relationships between the physical quantities that are involved in a particular phenomenon that one wishes to understand and characterize. It was used for the first time in this way in 1872 byLord Rayleigh, who was trying to understand why the sky is blue.[16] Rayleigh first published the technique in his 1877 bookThe Theory of Sound.[17]
The original meaning of the worddimension, in Fourier'sTheorie de la Chaleur, was the numerical value of the exponents of the base units. For example, acceleration was considered to have the dimension 1 with respect to the unit of length, and the dimension −2 with respect to the unit of time.[18] This was slightly changed by Maxwell, who said the dimensions of acceleration are T−2L, instead of just the exponents.[19]
What is the period ofoscillationT of a massm attached to an ideal linear spring with spring constantk suspended in gravity of strengthg? That period is the solution forT of some dimensionless equation in the variablesT,m,k, andg.The four quantities have the following dimensions:T [T];m [M];k [M/T2]; andg [L/T2]. From these we can form only one dimensionless product of powers of our chosen variables,G1 =T2k/m[T2 · M/T2 / M = 1], and puttingG1 =C for some dimensionless constantC gives the dimensionless equation sought. The dimensionless product of powers of variables is sometimes referred to as a dimensionless group of variables; here the term "group" means "collection" rather than mathematicalgroup. They are often calleddimensionless numbers as well.
The variableg does not occur in the group. It is easy to see that it is impossible to form a dimensionless product of powers that combinesg withk,m, andT, becauseg is the only quantity that involves the dimension L. This implies that in this problem theg is irrelevant. Dimensional analysis can sometimes yield strong statements about theirrelevance 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 ofg: 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:, for some dimensionless constantκ (equal to from the original dimensionless equation).
When faced with a case where dimensional analysis rejects a variable (g, here) that one intuitively expects to belong in a physical description of the situation, another possibility is that the rejected variable is in fact relevant, but that some other relevant variable has been omitted, which might combine with the rejected variable to form a dimensionless quantity. That is, however, not the case here.
When dimensional analysis yields only one dimensionless group, as here, there are no unknown functions, and the solution is said to be "complete" – although it still may involve unknown dimensionless constants, such asκ.
Consider the case of a vibrating wire oflengthℓ (L) vibrating with anamplitudeA (L). The wire has alinear densityρ (M/L) and is undertensions (LM/T2), and we want to know the energyE (L2M/T2) in the wire. Letπ1 andπ2 be two dimensionless products ofpowers of the variables chosen, given by
The linear density of the wire is not involved. The two groups found can be combined into an equivalent form as an equation
whereF is some unknown function, or, equivalently as
wheref is some other unknown function. Here the unknown function implies that our solution is now incomplete, but dimensional analysis has given us something that may not have been obvious: the energy is proportional to the first power of the tension. Barring further analytical analysis, we might proceed to experiments to discover the form for the unknown function f. But our experiments are simpler than in the absence of dimensional analysis. We'd perform none to verify that the energy is proportional to the tension. Or perhaps we might guess that the energy is proportional to ℓ, and so infer thatE =ℓs. The power of dimensional analysis as an aid to experiment and forming hypotheses becomes evident.
The power of dimensional analysis really becomes apparent when it is applied to situations, unlike those given above, that are more complicated, the set of variables involved are not apparent, and the underlying equations hopelessly complex. Consider, for example, a small pebble sitting on the bed of a river. If the river flows fast enough, it will actually raise the pebble and cause it to flow along with the water. At what critical velocity will this occur? Sorting out the guessed variables is not so easy as before. But dimensional analysis can be a powerful aid in understanding problems like this, and is usually the very first tool to be applied to complex problems where the underlying equations and constraints are poorly understood. In such cases, the answer may depend on adimensionless number such as theReynolds number, which may be interpreted by dimensional analysis.
Consider the case of a thin, solid, parallel-sided rotating disc of axial thicknesst (L) and radiusR (L). The disc has a densityρ (M/L3), rotates at an angular velocityω (T−1) and this leads to a stressS (T−2L−1M) in the material. There is a theoretical linear elastic solution, given by Lame, to this problem when the disc is thin relative to its radius, the faces of the disc are free to move axially, and the plane stress constitutive relations can be assumed to be valid. As the disc becomes thicker relative to the radius then the plane stress solution breaks down. If the disc is restrained axially on its free faces then a state of plane strain will occur. However, if this is not the case then the state of stress may only be determined though consideration of three-dimensional elasticity and there is no known theoretical solution for this case. An engineer might, therefore, be interested in establishing a relationship between the five variables. Dimensional analysis for this case leads to the following (5 − 3 = 2) non-dimensional groups:
Through the use of numerical experiments using, for example, thefinite element method, the nature of the relationship between the two non-dimensional groups can be obtained as shown in the figure. As this problem only involves two non-dimensional groups, the complete picture is provided in a single plot and this can be used as a design/assessment chart for rotating discs.[20]
The dimensions that can be formed from a given collection of basic physical dimensions, such as T, L, and M, form anabelian group: Theidentity is written as 1;[citation needed]L0 = 1, and the inverse of L is 1/L or L−1. L raised to any integer powerp is a member of the group, having an inverse of L−p or 1/Lp. The operation of the group is multiplication, having the usual rules for handling exponents (Ln × Lm = Ln+m). Physically, 1/L can be interpreted asreciprocal length, and 1/T as reciprocal time (seereciprocal second).
An abelian group is equivalent to amodule over the integers, with the dimensional symbolTiLjMk corresponding to the tuple(i,j,k). When physical measured quantities (be they like-dimensioned or unlike-dimensioned) are multiplied or divided by one other, their dimensional units are likewise multiplied or divided; this corresponds to addition or subtraction in the module. When measurable quantities are raised to an integer power, the same is done to the dimensional symbols attached to those quantities; this corresponds toscalar multiplication in the module.
A basis for such a module of dimensional symbols is called a set ofbase quantities, and all other vectors are called derived units. As in any module, one may choose differentbases, which yields different systems of units (e.g.,choosing whether the unit for charge is derived from the unit for current, or vice versa).
The group identity, the dimension of dimensionless quantities, corresponds to the origin in this module,(0, 0, 0).
In certain cases, one can define fractional dimensions, specifically by formally defining fractional powers of one-dimensional vector spaces, likeVL1/2.[21] However, it is not possible to take arbitrary fractional powers of units, due torepresentation-theoretic obstructions.[22]
One can work with vector spaces with given dimensions without needing to use units (corresponding to coordinate systems of the vector spaces). For example, given dimensionsM andL, one has the vector spacesVM andVL, and can defineVML :=VM ⊗VL as thetensor product. Similarly, the dual space can be interpreted as having "negative" dimensions.[23] This corresponds to the fact that under thenatural pairing between a vector space and its dual, the dimensions cancel, leaving adimensionless scalar.
The set of units of the physical quantities involved in a problem correspond to a set of vectors (or a matrix). Thenullity describes some number (e.g.,m) of ways in which these vectors can be combined to produce a zero vector. These correspond to producing (from the measurements) a number of dimensionless quantities,{π1, ..., πm}. (In fact these ways completely span the null subspace of another different space, of powers of the measurements.) Every possible way of multiplying (andexponentiating) together the measured quantities to produce something with the same unit as some derived quantityX can be expressed in the general form
Consequently, every possiblecommensurate equation for the physics of the system can be rewritten in the form
Knowing this restriction can be a powerful tool for obtaining new insight into the system.
The dimension of physical quantities of interest inmechanics can be expressed in terms of base dimensions T, L, and M – these form a 3-dimensional vector space. This is not the only valid choice of base dimensions, but it is the one most commonly used. For example, one might choose force, length and mass as the base dimensions (as some have done), with associated dimensions F, L, M; this corresponds to a different basis, and one may convert between these representations by achange of basis. The choice of the base set of dimensions is thus a convention, with the benefit of increased utility and familiarity. The choice of base dimensions is not entirely arbitrary, because they must form abasis: they mustspan the space, and belinearly independent.
For example, F, L, M form a set of fundamental dimensions because they form a basis that is equivalent to T, L, M: the former can be expressed as [F = LM/T2], L, M, while the latter can be expressed as [T = (LM/F)1/2], L, M.
On the other hand, length, velocity and time (T, L, V) do not form a set of base dimensions for mechanics, for two reasons:
Depending on the field of physics, it may be advantageous to choose one or another extended set of dimensional symbols. In electromagnetism, for example, it may be useful to use dimensions of T, L, M and Q, where Q represents the dimension ofelectric charge. Inthermodynamics, the base set of dimensions is often extended to include a dimension for temperature, Θ. In chemistry, theamount of substance (the number of molecules divided by theAvogadro constant, ≈6.02×1023 mol−1) is also defined as a base dimension, N.In the interaction ofrelativistic plasma with strong laser pulses, a dimensionlessrelativistic similarity parameter, connected with the symmetry properties of the collisionlessVlasov equation, is constructed from the plasma-, electron- and critical-densities in addition to the electromagnetic vector potential. The choice of the dimensions or even the number of dimensions to be used in different fields of physics is to some extent arbitrary, but consistency in use and ease of communications are common and necessary features.
Bridgman's theorem restricts the type of function that can be used to define a physical quantity from general (dimensionally compounded) quantities to only products of powers of the quantities, unless some of the independent quantities are algebraically combined to yield dimensionless groups, whose functions are grouped together in the dimensionless numeric multiplying factor.[24][25] This excludes polynomials of more than one term or transcendental functions not of that form.
Scalar arguments totranscendental functions such asexponential,trigonometric andlogarithmic functions, or toinhomogeneous polynomials, must bedimensionless quantities. (Note: this requirement is somewhat relaxed in Siano's orientational analysis described below, in which the square of certain dimensioned quantities are dimensionless.)
While most mathematical identities about dimensionless numbers translate in a straightforward manner to dimensional quantities, care must be taken with logarithms of ratios: the identitylog(a/b) = log a − log b, where the logarithm is taken in any base, holds for dimensionless numbersa andb, but it doesnot hold ifa andb are dimensional, because in this case the left-hand side is well-defined but the right-hand side is not.[26]
Similarly, while one can evaluatemonomials (xn) of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: forx2, the expression(3 m)2 = 9 m2 makes sense (as an area), while forx2 +x, the expression(3 m)2 + 3 m = 9 m2 + 3 m does not make sense.
However, polynomials of mixed degree can make sense if the coefficients are suitably chosen physical quantities that are not dimensionless. For example,
This is the height to which an object rises in time t if the acceleration ofgravity is 9.8metres per second per second and the initial upward speed is 500metres per second. It is not necessary fort to be inseconds. For example, supposet = 0.01 minutes. Then the first term would be
The value of a dimensional physical quantityZ is written as the product of aunit [Z] within the dimension and a dimensionless numerical value or numerical factor,n.[27]
When like-dimensioned quantities are added or subtracted or compared, it is convenient to express them in the same unit so that the numerical values of these quantities may be directly added or subtracted. But, in concept, there is no problem adding quantities of the same dimension expressed in different units. For example, 1 metre added to 1 foot is a length, but one cannot derive that length by simply adding 1 and 1. Aconversion factor, which is a ratio of like-dimensioned quantities and is equal to the dimensionless unity, is needed:
The factor 0.3048 m/ft is identical to the dimensionless 1, so multiplying by this conversion factor changes nothing. Then when adding two quantities of like dimension, but expressed in different units, the appropriate conversion factor, which is essentially the dimensionless 1, is used to convert the quantities to the same unit so that their numerical values can be added or subtracted.
Only in this manner is it meaningful to speak of adding like-dimensioned quantities of differing units.
Aquantity equation, also sometimes called acomplete equation, is an equation that remains valid independently of theunit of measurement used when expressing thephysical quantities.[28]
In contrast, in anumerical-value equation, just the numerical values of the quantities occur, without units. Therefore, it is only valid when each numerical values is referenced to a specific unit.
For example, a quantity equation fordisplacementd asspeeds multiplied by time differencet would be:
fors = 5 m/s, wheret andd may be expressed in any units,converted if necessary.In contrast, a corresponding numerical-value equation would be:
whereT is the numeric value oft when expressed in seconds andD is the numeric value ofd when expressed in metres.
Generally, the use of numerical-value equations is discouraged.[28]
The dimensionless constants that arise in the results obtained, such as theC in the Poiseuille's Law problem and theκ in the spring problems discussed above, come from a more detailed analysis of the underlying physics and often arise from integrating some differential equation. Dimensional analysis itself has little to say about these constants, but it is useful to know that they very often have a magnitude of order unity. This observation can allow one to sometimes make "back of the envelope" calculations about the phenomenon of interest, and therefore be able to more efficiently design experiments to measure it, or to judge whether it is important, etc.
Paradoxically, dimensional analysis can be a useful tool even if all the parameters in the underlying theory are dimensionless, e.g., lattice models such as theIsing model can be used to study phase transitions and critical phenomena. Such models can be formulated in a purely dimensionless way. As we approach the critical point closer and closer, the distance over which the variables in the lattice model are correlated (the so-called correlation length,χ) becomes larger and larger. Now, the correlation length is the relevant length scale related to critical phenomena, so one can, e.g., surmise on "dimensional grounds" that the non-analytical part of the free energy per lattice site should be~ 1/χd, whered is the dimension of the lattice.
It has been argued by some physicists, e.g.,Michael J. Duff,[4][29] that the laws of physics are inherently dimensionless. The fact that we have assigned incompatible dimensions to Length, Time and Mass is, according to this point of view, just a matter of convention, borne out of the fact that before the advent of modern physics, there was no way to relate mass, length, and time to each other. The three independent dimensionful constants:c,ħ, andG, in the fundamental equations of physics must then be seen as mere conversion factors to convert Mass, Time and Length into each other.
Just as in the case of critical properties of lattice models, one can recover the results of dimensional analysis in the appropriate scaling limit; e.g., dimensional analysis in mechanics can be derived by reinserting the constantsħ,c, andG (but we can now consider them to be dimensionless) and demanding that a nonsingular relation between quantities exists in the limitc → ∞,ħ → 0 andG → 0. In problems involving a gravitational field the latter limit should be taken such that the field stays finite.
Following are tables of commonly occurring expressions in physics, related to the dimensions of energy, momentum, and force.[30][31][32]
| Energy,E T−2L2M | Expression | Nomenclature |
|---|---|---|
| Mechanical | F =force,d =distance | |
| S =action,t = time,P =power | ||
| m =mass,v =velocity,p =momentum | ||
| L =angular momentum,I =moment of inertia,ω =angular velocity | ||
| Ideal gases | p = pressure,V = volume,T = temperature,N =amount of substance | |
| Waves | A =area ofwave front,I = waveintensity,t =time,S =Poynting vector | |
| Electromagnetic | q =electric charge,ϕ =electric potential (for changes this isvoltage) | |
| E =electric field,B =magnetic field, ε =permittivity,μ =permeability, V = 3dvolume | ||
| p =electric dipole moment,m = magnetic moment, A = area (bounded by a current loop),I =electric current in loop |
| Momentum,p T−1LM | Expression | Nomenclature |
|---|---|---|
| Mechanical | m = mass,v = velocity,F = force,t = time | |
| S = action,L = angular momentum,r =displacement | ||
| Thermal | =root mean square velocity,m = mass (of a molecule) | |
| Waves | ρ =density,V =volume,v =phase velocity | |
| Electromagnetic | A =magnetic vector potential |
| Force,F T−2LM | Expression | Nomenclature |
|---|---|---|
| Mechanical | m = mass,a = acceleration | |
| Thermal | S = entropy,T = temperature,r = displacement (seeentropic force) | |
| Electromagnetic | E = electric field,B = magnetic field,v = velocity,q = charge |
Dimensional correctness as part oftype checking has been studied since 1977.[33]Implementations for Ada[34] and C++[35] were described in 1985 and 1988.Kennedy's 1996 thesis describes an implementation inStandard ML,[36] and later inF#.[37] There are implementations forHaskell,[38]OCaml,[39] andRust,[40] Python,[41] and a code checker forFortran.[42][43]
Griffioen's 2019 thesis extended Kennedy'sHindley–Milner type system to support Hart's matrices.[44][45]McBride and Nordvall-Forsberg show how to usedependent types to extend type systems for units of measure.[46]
Mathematica 13.2 has a function for transformations with quantities named NondimensionalizationTransform that applies a nondimensionalization transform to an equation.[47] Mathematica also has a function to find the dimensions of a unit such as 1 J named UnitDimensions.[48] Mathematica also has a function that will find dimensionally equivalent combinations of a subset of physical quantities named DimensionalCombations.[49] Mathematica can also factor out certain dimension with UnitDimensions by specifying an argument to the function UnityDimensions.[50] For example, you can use UnityDimensions to factor out angles.[50] In addition to UnitDimensions, Mathematica can find the dimensions of a QuantityVariable with the function QuantityVariableDimensions.[51]
Some discussions of dimensional analysis implicitly describe all quantities as mathematical vectors. In mathematics scalars are considered a special case of vectors;[citation needed] 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: anorigin. 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:
This illustrates the subtle distinction betweenaffine quantities (ones modeled by anaffine space, such as position) andvector quantities (ones modeled by avector space, such as displacement).
Properly then, positions have dimension ofaffine length, while displacements have dimension ofvector length. To assign a number to anaffine unit, one must not only choose a unit of measurement, but also apoint of reference, while to assign a number to avector 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 ofabsolute zero is not the origin 0 in some scales. For absolute zero,
where the symbol ≘ meanscorresponds 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,
(Here °R refers to theRankine scale, not theRéaumur scale).Unit conversion for temperature differences is simply a matter of multiplying by, e.g., 1 °F / 1 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 K means the absolute temperature equal to −272.15 °C, or the temperature difference equal to 1 °C.
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 adirection. (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 aframe of reference.
This leads to theextensions discussed below, namely Huntley's directed dimensions and Siano's orientational analysis.
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 of the dimensional matrix.[52]
He introduced two approaches:
As an example of the usefulness of the first approach, suppose we wish to calculate thedistance a cannonball travels when fired with a vertical velocity component and a horizontal velocity component, assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are thenR, the distance travelled, with dimension L,,, both dimensioned as T−1L, andg the downward acceleration of gravity, with dimension T−2L.
With these four quantities, we may conclude that the equation for the rangeR may be written:
Or dimensionally
from which we may deduce that and, 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 will be dimensioned as T−1Lx, as T−1Ly,R as Lx andg as T−2Ly. The dimensional equation becomes:
and we may solve completely asa = 1,b = 1 andc = −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 also is often quite difficult to assign the L, Lx, Ly, Lz, 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.
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 onlyproportional to inertial mass, while not implicating inertial properties. No further restrictions are added to its definition.
For example, consider the derivation ofPoiseuille'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:
| Symbol | Variable | Dimension |
|---|---|---|
| mass flow rate | T−1M | |
| pressure gradient along the pipe | T−2L−2M | |
| ρ | density | L−3M |
| η | dynamic fluid viscosity | T−1L−1M |
| r | radius of the pipe | L |
There are three fundamental variables, so the above five equations will yield two independent dimensionless variables:
If we distinguish between inertial mass with dimension and quantity of matter with dimension, 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:
where now onlyC is an undetermined constant (found to be equal to by methods outside of dimensional analysis). This equation may be solved for the mass flow rate to yieldPoiseuille'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 dimensionamount of substance, with unitmole, 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.
Angles are, by convention, considered to be dimensionless quantities (although the wisdom of this is contested[53]) . As an example, consider again the projectile problem in which a point mass is launched from the origin(x,y) = (0, 0) at a speedv and angleθ above thex-axis, with the force of gravity directed along the negativey-axis. It is desired to find the rangeR, at which point the mass returns to thex-axis. Conventional analysis will yield the dimensionless variableπ =Rg/v2, but offers no insight into the relationship betweenR andθ.
Siano has suggested that the directed dimensions of Huntley be replaced by usingorientational symbols1x 1y 1z to denote vector directions, and an orientationless symbol 10.[54] Thus, Huntley's Lx becomes L1x with L specifying the dimension of length, and1x specifying the orientation. Siano further shows that the orientational symbols have an algebra of their own. Along with the requirement that1i−1 = 1i, the following multiplication table for the orientation symbols results:
The orientational symbols form a group (theKlein 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 of1z. For angles, consider an angleθ that lies in the z-plane. Form a right triangle in the z-plane withθ being one of the acute angles. The side of the right triangle adjacent to the angle then has an orientation1x and the side opposite has an orientation1y. Since (using~ to indicate orientational equivalence)tan(θ) =θ + ... ~ 1y/1x we conclude that an angle in the xy-plane must have an orientation1y/1x = 1z, which is not unreasonable. Analogous reasoning forces the conclusion thatsin(θ) has orientation1z whilecos(θ) has orientation 10. These are different, so one concludes (correctly), for example, that there are no solutions of physical equations that are of the forma cos(θ) +b sin(θ), wherea andb are real scalars. An expression such as is not dimensionally inconsistent since it is a special case of the sum of angles formula and should properly be written:
which for and yields. 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.
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 intonormal 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,θ, being in the xy-plane will thus have dimension1z and the range of the projectileR will be of the form:
Dimensional homogeneity will now correctly yielda = −1 andb = 2, and orientational homogeneity requires that. In other words, thatc must be an odd integer. In fact, the required function of theta will besin(θ)cos(θ) which is a series consisting of odd powers ofθ.
It is seen that the Taylor series ofsin(θ) andcos(θ) are orientationally homogeneous using the above multiplication table, while expressions likecos(θ) + sin(θ) andexp(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, theradian 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.
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.
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