Buckingham π theorem - Wikipedia

In engineering, applied mathematics, and physics, the Buckingham π theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's ... Buckinghamπtheorem FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Thisarticleneedsadditionalcitationsforverification.Pleasehelpimprovethisarticlebyaddingcitationstoreliablesources.Unsourcedmaterialmaybechallengedandremoved.Findsources: "Buckinghamπtheorem" – news ·newspapers ·books ·scholar ·JSTOR(April2022)(Learnhowandwhentoremovethistemplatemessage) Theoremindimensionalanalysis EdgarBuckinghamcirca1886 Inengineering,appliedmathematics,andphysics,theBuckinghamπtheoremisakeytheoremindimensionalanalysis.ItisaformalizationofRayleigh'smethodofdimensionalanalysis.Loosely,thetheoremstatesthatifthereisaphysicallymeaningfulequationinvolvingacertainnumbernofphysicalvariables,thentheoriginalequationcanberewrittenintermsofasetofp=n−kdimensionlessparametersπ1,π2,...,πpconstructedfromtheoriginalvariables.(Herekisthenumberofphysicaldimensionsinvolved;itisobtainedastherankofaparticularmatrix.) Thetheoremprovidesamethodforcomputingsetsofdimensionlessparametersfromthegivenvariables,ornondimensionalization,eveniftheformoftheequationisstillunknown. TheBuckinghamπtheoremindicatesthatvalidityofthelawsofphysicsdoesnotdependonaspecificunitsystem.Astatementofthistheoremisthatanyphysicallawcanbeexpressedasanidentityinvolvingonlydimensionlesscombinations(ratiosorproducts)ofthevariableslinkedbythelaw(forexample,pressureandvolumearelinkedbyBoyle'slaw–theyareinverselyproportional).Ifthedimensionlesscombinations'valueschangedwiththesystemsofunits,thentheequationwouldnotbeanidentity,andthetheoremwouldnothold. Contents 1History 2Statement 3Significance 4Proof 4.1Rescalingunits 4.2Formalproof 5Examples 5.1Speed 5.2Thesimplependulum 5.3Otherexamples 6Seealso 7References 7.1Notes 7.2Citations 7.3Bibliography 7.4Originalsources 8Externallinks History[edit] AlthoughnamedforEdgarBuckingham,theπtheoremwasfirstprovedbytheFrenchmathematicianJosephBertrand[1]in1878.Bertrandconsideredonlyspecialcasesofproblemsfromelectrodynamicsandheatconduction,buthisarticlecontains,indistinctterms,allthebasicideasofthemodernproofofthetheoremandclearlyindicatesthetheorem'sutilityformodellingphysicalphenomena.Thetechniqueofusingthetheorem("themethodofdimensions")becamewidelyknownduetotheworksofRayleigh.Thefirstapplicationoftheπtheoreminthegeneralcase[note1]tothedependenceofpressuredropinapipeupongoverningparametersprobablydatesbackto1892,[2]aheuristicproofwiththeuseofseriesexpansions,to1894.[3] FormalgeneralizationoftheπtheoremforthecaseofarbitrarilymanyquantitieswasgivenfirstbyA.Vaschy [fr]in1892,[4]thenin1911—apparentlyindependently—bybothA.Federman[5]andD.Riabouchinsky,[6]andagainin1914byBuckingham.[7]ItwasBuckingham'sarticlethatintroducedtheuseofthesymbol" π i {\displaystyle\pi_{i}} "forthedimensionlessvariables(orparameters),andthisisthesourceofthetheorem'sname. Statement[edit] Moreformally,thenumber p {\displaystylep} ofdimensionlesstermsthatcanbeformedisequaltothenullityofthedimensionalmatrix,and k {\displaystylek} istherank.Forexperimentalpurposes,differentsystemsthatsharethesamedescriptionintermsofthesedimensionlessnumbersareequivalent. Inmathematicalterms,ifwehaveaphysicallymeaningfulequationsuchas f ( q 1 , q 2 , … , q n ) = 0 , {\displaystylef(q_{1},q_{2},\ldots,q_{n})=0,} where q 1 , … , q n {\displaystyleq_{1},\ldots,q_{n}} arethe n {\displaystylen} independentphysicalvariables,andthereisamaximaldimensionallyindependentsubsetofsize k {\displaystylek} ,[note2]thentheaboveequationcanberestatedas F ( π 1 , π 2 , … , π p ) = 0 , {\displaystyleF(\pi_{1},\pi_{2},\ldots,\pi_{p})=0,} where π 1 , … , π p {\displaystyle\pi_{1},\ldots,\pi_{p}} aredimensionlessparametersconstructedfromthe q i {\displaystyleq_{i}} by p = n − k {\displaystylep=n-k} dimensionlessequations—theso-calledPigroups—oftheform π i = q 1 a 1 q 2 a 2 ⋯ q n a n , {\displaystyle\pi_{i}=q_{1}^{a_{1}}\,q_{2}^{a_{2}}\cdotsq_{n}^{a_{n}},} wheretheexponents a i {\displaystylea_{i}} arerationalnumbers.(Theycanalwaysbetakentobeintegersbyredefining π i {\displaystyle\pi_{i}} asbeingraisedtoapowerthatclearsalldenominators.)Ifthereare ℓ {\displaystyle\ell} fundamentalunitsinplay,then p ≥ n − ℓ {\displaystylep\geqn-\ell} . Significance[edit] TheBuckinghamπtheoremprovidesamethodforcomputingsetsofdimensionlessparametersfromgivenvariables,eveniftheformoftheequationremainsunknown.However,thechoiceofdimensionlessparametersisnotunique;Buckingham'stheoremonlyprovidesawayofgeneratingsetsofdimensionlessparametersanddoesnotindicatethemost"physicallymeaningful". Twosystemsforwhichtheseparameterscoincidearecalledsimilar(aswithsimilartriangles,theydifferonlyinscale);theyareequivalentforthepurposesoftheequation,andtheexperimentalistwhowantstodeterminetheformoftheequationcanchoosethemostconvenientone.Mostimportantly,Buckingham'stheoremdescribestherelationbetweenthenumberofvariablesandfundamentaldimensions. Proof[edit] Forsimplicity,itwillbeassumedthatthespaceoffundamentalandderivedphysicalunitsformsavectorspaceovertherealnumbers,withthefundamentalunitsasbasisvectors,andwithmultiplicationofphysicalunitsasthe"vectoraddition"operation,andraisingtopowersasthe"scalarmultiplication"operation: representadimensionalvariableasthesetofexponentsneededforthefundamentalunits(withapowerofzeroiftheparticularfundamentalunitisnotpresent).Forinstance,thestandardgravity g {\displaystyleg} hasunitsof D / T 2 = D 1 T − 2 {\displaystyleD/T^{2}=D^{1}T^{-2}} (distanceovertimesquared),soitisrepresentedasthevector ( 1 , − 2 ) {\displaystyle(1,-2)} withrespecttothebasisoffundamentalunits(distance,time).Wecouldalsorequirethatexponentsofthefundamentalunitsberationalnumbersandmodifytheproofaccordingly,inwhichcasetheexponentsinthepigroupscanalwaysbetakenasrationalnumbersorevenintegers. Rescalingunits[edit] Supposewehavequantities q 1 , q 2 , … , q n {\displaystyleq_{1},q_{2},\dots,q_{n}} ,wheretheunitsof q i {\displaystyleq_{i}} containlengthraisedtothepower c i {\displaystylec_{i}} .Ifweoriginallymeasurelengthinmetersbutlaterswitchtocentimeters,thenthenumericalvalueof q i {\displaystyleq_{i}} wouldberescaledbyafactorof 100 c i {\displaystyle100^{c_{i}}} .Anyphysicallymeaningfullawshouldbeinvariantunderanarbitraryrescalingofeveryfundamentalunit;thisisthefactthatthepitheoremhingeson. Formalproof[edit] Givenasystemof n {\displaystylen} dimensionalvariables q 1 , … , q n {\displaystyleq_{1},\ldots,q_{n}} (withphysicaldimensions)in ℓ {\displaystyle\ell} fundamental(basis)dimensions,thedimensionalmatrixisthe ℓ × n {\displaystyle\ell\timesn} matrix M {\displaystyleM} whose ℓ {\displaystyle\ell} rowsarethefundamentaldimensionsandwhose n {\displaystylen} columnsarethedimensionsofthevariables:the ( i , j ) {\displaystyle(i,j)} thentry(where 1 ≤ i ≤ ℓ {\displaystyle1\leqi\leq\ell} and 1 ≤ j ≤ n {\displaystyle1\leqj\leqn} )isthepowerofthe i {\displaystylei} thfundamentaldimensioninthe j {\displaystylej} thvariable. Thematrixcanbeinterpretedastakinginacombinationofthedimensionsofthevariablequantitiesandgivingoutthedimensionsofthisproductinfundamentaldimensions.Sothe ℓ × 1 {\displaystyle\ell\times1} (column)vectorthatresultsfromthemultiplication M [ a 1 ⋮ a n ] {\displaystyleM{\begin{bmatrix}a_{1}\\\vdots\\a_{n}\end{bmatrix}}} consistsoftheunitsof q 1 a 1 q 2 a 2 ⋯ q n a n {\displaystyleq_{1}^{a_{1}}\,q_{2}^{a_{2}}\cdotsq_{n}^{a_{n}}} intermsofthe ℓ {\displaystyle\ell} fundamentalindependent(basis)units.[note3] Ifwerescalethe i {\displaystylei} thfundamentalunitbyafactorof α i {\displaystyle\alpha_{i}} ,then q j {\displaystyleq_{j}} getsrescaledby α 1 − m 1 j α 2 − m 2 j ⋯ α ℓ − m ℓ j {\displaystyle\alpha_{1}^{-m_{1j}}\,\alpha_{2}^{-m_{2j}}\cdots\alpha_{\ell}^{-m_{\ellj}}} ,where m i j {\displaystylem_{ij}} isthe ( i , j ) {\displaystyle(i,j)} thentryofthedimensionalmatrix.Inordertoconvertthisintoalinearalgebraproblem,wetakelogarithms(thebaseisirrelevant),yielding [ log ⁡ q 1 ⋮ log ⁡ q n ] ↦ [ log ⁡ q 1 ⋮ log ⁡ q n ] − M T [ log ⁡ α 1 ⋮ log ⁡ α ℓ ] , {\displaystyle{\begin{bmatrix}\log{q_{1}}\\\vdots\\\log{q_{n}}\end{bmatrix}}\mapsto{\begin{bmatrix}\log{q_{1}}\\\vdots\\\log{q_{n}}\end{bmatrix}}-M^{\operatorname{T}}{\begin{bmatrix}\log{\alpha_{1}}\\\vdots\\\log{\alpha_{\ell}}\end{bmatrix}},} whichisanactionof R ℓ {\displaystyle\mathbb{R}^{\ell}} on R n {\displaystyle\mathbb{R}^{n}} .Wedefineaphysicallawtobeanarbitraryfunction f : ( R + ) n → R {\displaystylef\colon(\mathbb{R}^{+})^{n}\to\mathbb{R}} suchthat ( q 1 , q 2 , … , q n ) {\displaystyle(q_{1},q_{2},\dots,q_{n})} isapermissiblesetofvaluesforthephysicalsystemwhen f ( q 1 , q 2 , … , q n ) = 0 {\displaystylef(q_{1},q_{2},\dots,q_{n})=0} .Wefurtherrequire f {\displaystylef} tobeinvariantunderthisaction.Henceitdescendstoafunction F : R n / im ⁡ M T → R {\displaystyleF\colon\mathbb{R}^{n}/\operatorname{im}{M^{\operatorname{T}}}\to\mathbb{R}} .Allthatremainsistoexhibitanisomorphismbetween R n / im ⁡ M T {\displaystyle\mathbb{R}^{n}/\operatorname{im}{M^{\operatorname{T}}}} and R p {\displaystyle\mathbb{R}^{p}} ,the(log)spaceofpigroups ( log ⁡ π 1 , log ⁡ π 2 , … , log ⁡ π p ) {\displaystyle(\log{\pi_{1}},\log{\pi_{2}},\dots,\log{\pi_{p}})} . Weconstructan n × p {\displaystylen\timesp} matrix K {\displaystyleK} whosecolumnsareabasisfor ker ⁡ M {\displaystyle\ker{M}} .Ittellsushowtoembed R p {\displaystyle\mathbb{R}^{p}} into R n {\displaystyle\mathbb{R}^{n}} asthekernelof M {\displaystyleM} .Thatis,wehaveashortexactsequence 0 → R p →   K   R n →   M   R ℓ → 0. {\displaystyle0\to\mathbb{R}^{p}\xrightarrow{\K\}\mathbb{R}^{n}\xrightarrow{\M\}\mathbb{R}^{\ell}\to0.} Takingtranposesyieldsanothershortexactsequence 0 → R ℓ →   M T   R n →   K T   R p → 0. {\displaystyle0\to\mathbb{R}^{\ell}\xrightarrow{\M^{\operatorname{T}}\}\mathbb{R}^{n}\xrightarrow{\K^{\operatorname{T}}\}\mathbb{R}^{p}\to0.} Thefirstisomorphismtheoremproducesthedesiredisomorphism,whichsendsthecoset v + M T R ℓ {\displaystylev+M^{\operatorname{T}}\mathbb{R}^{\ell}} to K T v {\displaystyleK^{\operatorname{T}}v} .Thiscorrespondstorewritingthetuple ( log ⁡ q 1 , log ⁡ q 2 , … , log ⁡ q n ) {\displaystyle(\log{q_{1}},\log{q_{2}},\dots,\log{q_{n}})} intothepigroups ( log ⁡ π 1 , log ⁡ π 2 , … , log ⁡ π p ) {\displaystyle(\log{\pi_{1}},\log{\pi_{2}},\dots,\log{\pi_{p}})} comingfromthecolumnsof K {\displaystyleK} . TheInternationalSystemofUnitsdefinessevenbaseunits,whicharetheampere,kelvin,second,metre,kilogram,candelaandmole.Itissometimesadvantageoustointroduceadditionalbaseunitsandtechniquestorefinethetechniqueofdimensionalanalysis.(Seeorientationalanalysisandreference.[8]) Examples[edit] Speed[edit] Thisexampleiselementarybutservestodemonstratetheprocedure. Supposeacarisdrivingat100 km/h;howlongdoesittaketogo200 km? Thisquestionconsiders n = 3 {\displaystylen=3} dimensionedvariables:distance d , {\displaystyled,} time t , {\displaystylet,} andspeed v , {\displaystylev,} andweareseekingsomelawoftheform t = Duration ⁡ ( v , d ) . {\displaystylet=\operatorname{Duration}(v,d).} Anytwoofthesevariablesaredimensionallyindependent,butthethreetakentogetherarenot.Thusthereis p = n − k = 3 − 2 = 1 {\displaystylep=n-k=3-2=1} dimensionlessquantity. Thedimensionalmatrixis M = [ 1 0 1 0 1 − 1 ] {\displaystyleM={\begin{bmatrix}1&0&\;\;\;1\\0&1&-1\end{bmatrix}}} inwhichtherowscorrespondtothebasisdimensions D {\displaystyleD} and T , {\displaystyleT,} andthecolumnstotheconsidereddimensions D , T ,  and  V , {\displaystyleD,T,{\text{and}}V,} wherethelatterstandsforthespeeddimension.Theelementsofthematrixcorrespondtothepowerstowhichtherespectivedimensionsaretoberaised.Forinstance,thethirdcolumn ( 1 , − 1 ) , {\displaystyle(1,-1),} statesthat V = D 0 T 0 V 1 , {\displaystyleV=D^{0}T^{0}V^{1},} representedbythecolumnvector v = [ 0 , 0 , 1 ] , {\displaystyle\mathbf{v}=[0,0,1],} isexpressibleintermsofthebasisdimensionsas V = D 1 T − 1 = D / T , {\displaystyleV=D^{1}T^{-1}=D/T,} since M v = [ 1 , − 1 ] . {\displaystyleM\mathbf{v}=[1,-1].} Foradimensionlessconstant π = D a 1 T a 2 V a 3 , {\displaystyle\pi=D^{a_{1}}T^{a_{2}}V^{a_{3}},} wearelookingforvectors a = [ a 1 , a 2 , a 3 ] {\displaystyle\mathbf{a}=[a_{1},a_{2},a_{3}]} suchthatthematrix-vectorproduct M a {\displaystyleM\mathbf{a}} equalsthezerovector [ 0 , 0 ] . {\displaystyle[0,0].} Inlinearalgebra,thesetofvectorswiththispropertyisknownasthekernel(ornullspace)of(thelinearmaprepresentedby)thedimensionalmatrix.Inthisparticularcaseitskernelisone-dimensional.Thedimensionalmatrixaswrittenaboveisinreducedrowechelonform,soonecanreadoffanon-zerokernelvectortowithinamultiplicativeconstant: a = [ − 1 1 1 ] . {\displaystyle\mathbf{a}={\begin{bmatrix}-1\\\;\;\;1\\\;\;\;1\\\end{bmatrix}}.} Ifthedimensionalmatrixwerenotalreadyreduced,onecouldperformGauss–Jordaneliminationonthedimensionalmatrixtomoreeasilydeterminethekernel.Itfollowsthatthedimensionlessconstant,replacingthedimensionsbythecorrespondingdimensionedvariables,maybewritten: π = d − 1 t 1 v 1 = t v / d . {\displaystyle\pi=d^{-1}t^{1}v^{1}=tv/d.} Sincethekernelisonlydefinedtowithinamultiplicativeconstant,theabovedimensionlessconstantraisedtoanyarbitrarypoweryieldsanother(equivalent)dimensionlessconstant. Dimensionalanalysishasthusprovidedageneralequationrelatingthethreephysicalvariables: F ( π ) = 0 , {\displaystyleF(\pi)=0,} or,letting C {\displaystyleC} denoteazerooffunction F , {\displaystyleF,} π = C , {\displaystyle\pi=C,} whichcanbewritteninthedesiredform(whichrecallwas t = Duration ⁡ ( v , d ) {\displaystylet=\operatorname{Duration}(v,d)} )as t = C d v . {\displaystylet=C{\frac{d}{v}}.} Theactualrelationshipbetweenthethreevariablesissimply d = v t . {\displaystyled=vt.} Inotherwords,inthiscase F {\displaystyleF} hasonephysicallyrelevantroot,anditisunity.Thefactthatonlyasinglevalueof C {\displaystyleC} willdoandthatitisequalto1isnotrevealedbythetechniqueofdimensionalanalysis. Thesimplependulum[edit] Wewishtodeterminetheperiod T {\displaystyleT} ofsmalloscillationsinasimplependulum.Itwillbeassumedthatitisafunctionofthelength L , {\displaystyleL,} themass M , {\displaystyleM,} andtheaccelerationduetogravityonthesurfaceoftheEarth g , {\displaystyleg,} whichhasdimensionsoflengthdividedbytimesquared.Themodelisoftheform f ( T , M , L , g ) = 0. {\displaystylef(T,M,L,g)=0.} (Notethatitiswrittenasarelation,notasafunction: T {\displaystyleT} isnotwrittenhereasafunctionof M , L ,  and  g . {\displaystyleM,L,{\text{and}}g.} ) Period,mass,andlengtharedimensionallyindependent,butaccelerationcanbeexpressedintermsoftimeandlength,whichmeansthefourvariablestakentogetherarenotdimensionallyindependent.Thusweneedonly p = n − k = 4 − 3 = 1 {\displaystylep=n-k=4-3=1} dimensionlessparameter,denotedby π , {\displaystyle\pi,} andthemodelcanbere-expressedas F ( π ) = 0 , {\displaystyleF(\pi)=0,} where π {\displaystyle\pi} isgivenby π = T a 1 M a 2 L a 3 g a 4 {\displaystyle\pi=T^{a_{1}}M^{a_{2}}L^{a_{3}}g^{a_{4}}} forsomevaluesof a 1 , a 2 , a 3 , a 4 . {\displaystylea_{1},a_{2},a_{3},a_{4}.} Thedimensionsofthedimensionalquantitiesare: T = t , M = m , L = ℓ , g = ℓ / t 2 . {\displaystyleT=t,M=m,L=\ell,g=\ell/t^{2}.} Thedimensionalmatrixis: M = [ 1 0 0 − 2 0 1 0 0 0 0 1 1 ] . {\displaystyle\mathbf{M}={\begin{bmatrix}1&0&0&-2\\0&1&0&0\\0&0&1&1\end{bmatrix}}.} (Therowscorrespondtothedimensions t , m , {\displaystylet,m,} and ℓ , {\displaystyle\ell,} andthecolumnstothedimensionalvariables T , M , L ,  and  g . {\displaystyleT,M,L,{\text{and}}g.} Forinstance,the4thcolumn, ( − 2 , 0 , 1 ) , {\displaystyle(-2,0,1),} statesthatthe g {\displaystyleg} variablehasdimensionsof t − 2 m 0 ℓ 1 . {\displaystylet^{-2}m^{0}\ell^{1}.} ) Wearelookingforakernelvector a = [ a 1 , a 2 , a 3 , a 4 ] {\displaystylea=\left[a_{1},a_{2},a_{3},a_{4}\right]} suchthatthematrixproductof M {\displaystyle\mathbf{M}} on a {\displaystylea} yieldsthezerovector [ 0 , 0 , 0 ] . {\displaystyle[0,0,0].} Thedimensionalmatrixaswrittenaboveisinreducedrowechelonform,soonecanreadoffakernelvectorwithinamultiplicativeconstant: a = [ 2 0 − 1 1 ] . {\displaystylea={\begin{bmatrix}2\\0\\-1\\1\end{bmatrix}}.} Wereitnotalreadyreduced,onecouldperformGauss–Jordaneliminationonthedimensionalmatrixtomoreeasilydeterminethekernel.Itfollowsthatthedimensionlessconstantmaybewritten: π = T 2 M 0 L − 1 g 1 = g T 2 / L . {\displaystyle{\begin{aligned}\pi&=T^{2}M^{0}L^{-1}g^{1}\\&=gT^{2}/L\end{aligned}}.} Infundamentalterms: π = ( t ) 2 ( m ) 0 ( ℓ ) − 1 ( ℓ / t 2 ) 1 = 1 , {\displaystyle\pi=(t)^{2}(m)^{0}(\ell)^{-1}\left(\ell/t^{2}\right)^{1}=1,} whichisdimensionless.Sincethekernelisonlydefinedtowithinamultiplicativeconstant,iftheabovedimensionlessconstantisraisedtoanyarbitrarypower,itwillyieldanotherequivalentdimensionlessconstant. Inthisexample,threeofthefourdimensionalquantitiesarefundamentalunits,sothelast(whichis g {\displaystyleg} )mustbeacombinationoftheprevious. Notethatif a 2 {\displaystylea_{2}} (thecoefficientof M {\displaystyleM} )hadbeennon-zerothentherewouldbenowaytocancelthe M {\displaystyleM} value;therefore a 2 {\displaystylea_{2}} mustbezero.Dimensionalanalysishasallowedustoconcludethattheperiodofthependulumisnotafunctionofitsmass M . {\displaystyleM.} (Inthe3Dspaceofpowersofmass,time,anddistance,wecansaythatthevectorformassislinearlyindependentfromthevectorsforthethreeothervariables.Uptoascalingfactor, g → + 2 T → − L → {\displaystyle{\vec{g}}+2{\vec{T}}-{\vec{L}}} istheonlynontrivialwaytoconstructavectorofadimensionlessparameter.) Themodelcannowbeexpressedas: F ( g T 2 / L ) = 0. {\displaystyleF\left(gT^{2}/L\right)=0.} Thenthisimpliesthat g T 2 / L = C i {\displaystylegT^{2}/L=C_{i}} forsomezero C i {\displaystyleC_{i}} ofthefunction F . {\displaystyleF.} Ifthereisonlyonezero,callit C , {\displaystyleC,} then g T 2 / L = C . {\displaystylegT^{2}/L=C.} Itrequiresmorephysicalinsightoranexperimenttoshowthatthereisindeedonlyonezeroandthattheconstantisinfactgivenby C = 4 π 2 . {\displaystyleC=4\pi^{2}.} Forlargeoscillationsofapendulum,theanalysisiscomplicatedbyanadditionaldimensionlessparameter,themaximumswingangle.Theaboveanalysisisagoodapproximationastheangleapproacheszero. Otherexamples[edit] Anexampleofdimensionalanalysiscanbefoundforthecaseofthemechanicsofathin,solidandparallel-sidedrotatingdisc.Therearefivevariablesinvolvedwhichreducetotwonon-dimensionalgroups.Therelationshipbetweenthesecanbedeterminedbynumericalexperimentusing,forexample,thefiniteelementmethod.[9] Thetheoremhasalsobeenusedinfieldsotherthanphysics,forinstanceinsportsscience.[10] Seealso[edit] Mathematicsportal Physicsportal Blastwave Dimensionlessquantity Naturalunits Similitude(model) Reynoldsnumber References[edit] Notes[edit] ^Wheninapplyingtheπ–theoremtherearisesanarbitraryfunctionofdimensionlessnumbers. ^Adimensionallyindependentsetofvariablesisoneforwhichtheonlyexponents q 1 a 1 q 2 a 2 ⋯ q k a k {\displaystyleq_{1}^{a_{1}}\,q_{2}^{a_{2}}\cdotsq_{k}^{a_{k}}} yieldingadimensionlessquantityare a 1 = a 2 = ⋯ = 0 {\displaystylea_{1}=a_{2}=\cdots=0} .Thisispreciselythenotionoflinearindependence. ^Ifthesebasisunitsare b 1 , … , b ℓ {\displaystyleb_{1},\ldots,b_{\ell}} andif q i = m 1 i b 1 + ⋯ + m ℓ i b k {\displaystyleq_{i}=m_{1i}b_{1}+\cdots+m_{\elli}b_{k}} forevery 1 ≤ i ≤ n {\displaystyle1\leqi\leqn} then M = [ m 11 ⋯ m 1 i ⋯ m 1 n ⋮ ⋮ ⋮ m ℓ 1 ⋯ m ℓ i ⋯ m ℓ n ] {\displaystyleM={\begin{bmatrix}m_{11}&\cdots&m_{1i}&\cdots&m_{1n}\\\vdots&&\vdots&&\vdots\\m_{\ell1}&\cdots&m_{\elli}&\cdots&m_{\elln}\\\end{bmatrix}}} sothatforinstance,theunitsof q 1 {\displaystyleq_{1}} intermsofthesebasisunitsare M ( [ 1   0   ⋯   0 ] T ) = [ m 11 ⋮ m ℓ 1 ] . {\displaystyleM\left(\left[1\0\\cdots\0\right]^{\operatorname{T}}\right)={\begin{bmatrix}m_{11}\\\vdots\\m_{\ell1}\\\end{bmatrix}}.} Foraconcreteexample,supposethatthe ℓ = 2 {\displaystyle\ell=2} fundamentalunitsaremeters b 1 = m {\displaystyleb_{1}=m} andseconds b 2 = s , {\displaystyleb_{2}=s,} andthatthereare n = 3 {\displaystylen=3} dimensionalvariables: q 1 = m / s 2 , q 2 = 1 / m , q 3 = s / m . {\displaystyleq_{1}=m/s^{2},q_{2}=1/m,q_{3}=s/m.} Bydefinitionofvectoradditionandscalarmultiplicationin V , {\displaystyleV,} q 1 = m s − 2 = 1 m + ( − 2 ) s , q 2 = m − 1 = ( − 1 ) m + 0 s , q 3 = m − 1 s = ( − 1 ) m + 1 s {\displaystyleq_{1}=ms^{-2}=1m+(-2)s,\quadq_{2}=m^{-1}=(-1)m+0s,\quadq_{3}=m^{-1}s=(-1)m+1s} sothat M = [ m 11 m 12 m 13 m 21 m 22 m 23 ] = [ 1 − 1 − 1 − 2 0 1 ] . {\displaystyleM={\begin{bmatrix}m_{11}&m_{12}&m_{13}\\m_{21}&m_{22}&m_{23}\\\end{bmatrix}}={\begin{bmatrix}1&-1&-1\\-2&0&1\\\end{bmatrix}}.} Bydefinition,thedimensionlessunitsarethoseoftheform m 0 s 0 , {\displaystylem^{0}s^{0},} whichareexactlythevectorsin ker ⁡ M = span ⁡ { [ 1 , − 1 , 2 ] T } = { ( q 1 − q 2 + 2 q 3 ) s : s ∈ Q } . {\displaystyle\kerM=\operatorname{span}\left\{[1,-1,2]^{\operatorname{T}}\right\}=\left\{\left(q_{1}-q_{2}+2q_{3}\right)^{s}:s\in\mathbb{Q}\right\}.} Thiscanbeverifiedbyadirectcomputation: q 1 − q 2 + 2 q 3 = ( m s − 2 ) 1 + ( m − 1 ) − 1 + ( s m − 1 ) 2 = m 1 s − 2 + m 1 + m − 2 s 2 = m 1 + 1 + ( − 2 ) s − 2 + 0 + 2 = m 0 s 0 , {\displaystyleq_{1}-q_{2}+2q_{3}=\left(ms^{-2}\right)^{1}+\left(m^{-1}\right)^{-1}+\left(sm^{-1}\right)^{2}=m^{1}s^{-2}+m^{1}+m^{-2}s^{2}=m^{1+1+(-2)}s^{-2+0+2}=m^{0}s^{0},} whichisindeeddimensionless. Consequently,ifsomephysicallawstatesthat q 1 , q 2 , q 3 {\displaystyleq_{1},q_{2},q_{3}} arenecessarilyrelatedbya(presumablyunknown)equationoftheform f ( q 1 , q 2 , q 3 ) = 0 {\displaystylef\left(q_{1},q_{2},q_{3}\right)=0} forsome(unknown)function f {\displaystylef} with domain ⁡ f ⊆ R 3 {\displaystyle\operatorname{domain}f\subseteq\mathbb{R}^{3}} (thatis,thetuple ( q 1 , q 2 , q 3 ) {\displaystyle\left(q_{1},q_{2},q_{3}\right)} isnecessarilyazeroof f {\displaystylef} ),thenthereexistssome(alsounknown)function F : R 1 → R {\displaystyleF:\mathbb{R}^{1}\to\mathbb{R}} thatdependsononly p = 3 − 2 = 1 {\displaystylep=3-2=1} variable,thedimensionlessvariable π 1 := q 1 − q 2 + 2 q 3 = q 1 q 3 2 / q 2 {\displaystyle\pi_{1}:=q_{1}-q_{2}+2q_{3}=q_{1}q_{3}^{2}/q_{2}} (oranynon-zerorationalpower π ^ 1 := π 1 s {\displaystyle{\hat{\pi}}_{1}:=\pi_{1}^{s}} of π 1 , {\displaystyle\pi_{1},} where 0 ≠ s ∈ Q {\displaystyle0\neqs\in\mathbb{Q}} ),suchthat F ( π 1 ) = 0 {\displaystyleF\left(\pi_{1}\right)=0} holds(if π ^ 1 := π 1 s {\displaystyle{\hat{\pi}}_{1}:=\pi_{1}^{s}} isusedinsteadof π 1 {\displaystyle\pi_{1}} then F {\displaystyleF} canbereplacedwith F ^ ( x ) := F ( x 1 / s ) {\displaystyle{\hat{F}}(x):=F\left(x^{1/s}\right)} andonceagain F ^ ( π ^ 1 ) = 0 {\displaystyle{\hat{F}}\left({\hat{\pi}}_{1}\right)=0} holds). Thusintermsoftheoriginalvariables, F ( q 1 q 3 2 / q 2 ) = 0 {\displaystyleF\left(q_{1}q_{3}^{2}/q_{2}\right)=0} musthold(alternatively,ifusing π ^ 1 := π 1 1 / 2 = π 1 {\displaystyle{\hat{\pi}}_{1}:=\pi_{1}^{1/2}={\sqrt{\pi_{1}}}} forinstance,then F ^ ( q 1 q 3 2 / q 2 ) = 0 {\displaystyle{\hat{F}}\left({\sqrt{q_{1}q_{3}^{2}/q_{2}}}\right)=0} musthold). Inotherwords,theBuckinghamπtheoremimpliesthat q 1 q 3 2 / q 2 ∈ F − 1 ( 0 ) , {\displaystyleq_{1}q_{3}^{2}/q_{2}\inF^{-1}(0),} sothatifithappenstobethecasethatthis F {\displaystyleF} hasexactlyonezero,callit C , {\displaystyleC,} thentheequation q 1 q 3 2 / q 2 = C {\displaystyleq_{1}q_{3}^{2}/q_{2}=C} willnecessarilyhold(thetheoremdoesnotgiveinformationaboutwhattheexactvalueoftheconstant C {\displaystyleC} willbe,nordoesitguaranteethat F {\displaystyleF} hasexactlyonezero). Citations[edit] ^Bertrand,J.(1878)."Surl'homogénéitédanslesformulesdephysique".ComptesRendus.86(15):916–920. ^Rayleigh(1892)."Onthequestionofthestabilityoftheflowofliquids".PhilosophicalMagazine.34(206):59–70.doi:10.1080/14786449208620167. ^Strutt,JohnWilliam(1896).TheTheoryofSound.Vol. II(2nd ed.).Macmillan. ^QuotesfromVaschy'sarticlewithhisstatementofthepi–theoremcanbefoundin:Macagno,E.O.(1971)."Historico-criticalreviewofdimensionalanalysis".JournaloftheFranklinInstitute.292(6):391–402.doi:10.1016/0016-0032(71)90160-8. ^Федерман,А.(1911)."Онекоторыхобщихметодахинтегрированияуравненийсчастнымипроизводнымипервогопорядка".ИзвестияСанкт-ПетербургскогополитехническогоинститутаимператораПетраВеликого.Отделтехники,естествознанияиматематики.16(1):97–155.(FedermanA.,Onsomegeneralmethodsofintegrationoffirst-orderpartialdifferentialequations,ProceedingsoftheSaint-Petersburgpolytechnicinstitute.Sectionoftechnics,naturalscience,andmathematics) ^Riabouchinsky,D.(1911)."Мéthodedesvariablesdedimensionzéroetsonapplicationenaérodynamique".L'Aérophile:407–408. ^Buckingham1914. ^Schlick,R.;LeSergent,T.(2006)."CheckingSCADEModelsforCorrectUsageofPhysicalUnits".ComputerSafety,Reliability,andSecurity.LectureNotesinComputerScience.Berlin:Springer.4166:358–371.doi:10.1007/11875567_27.ISBN 978-3-540-45762-6. ^Ramsay,Angus."DimensionalAnalysisandNumericalExperimentsforaRotatingDisc".RamsayMaunderAssociates.Retrieved15April2017. ^Blondeau,J.(2020)."Theinfluenceoffieldsize,goalsizeandnumberofplayersontheaveragenumberofgoalsscoredpergameinvariantsoffootballandhockey:thePi-theoremappliedtoteamsports".JournalofQuantitativeAnalysisinSports.17(2):145–154.doi:10.1515/jqas-2020-0009.S2CID 224929098. Bibliography[edit] Hanche-Olsen,Harald(2004)."Buckingham'spi-theorem"(PDF).NTNU.RetrievedApril9,2007. Hart,GeorgeW.(1995).MultidimensionalAnalysis:AlgebrasandSystemsforScienceandEngineering.Springer.ISBN 978-0-387-94417-3. Kline,StephenJ.(1986).SimilitudeandApproximationTheory.Springer.ISBN 978-0-387-16518-9. Hartke,Jan-David(2019)."OnBuckingham'sΠ-theorem".arXiv:1912.08744. Wan,FredericY.M.(1989).MathematicalModelsandtheirAnalysis.Harper&Row.ISBN 978-0-06-046902-3. Vignaux,G.A.(1991)."Dimensionalanalysisindatamodelling"(PDF).VictoriaUniversityofWellington.RetrievedDecember15,2005. Sheppard,Mike(2008)."SystematicSearchforExpressionsofDimensionlessConstantsusingtheNISTdatabaseofPhysicalConstants".Archivedfromtheoriginalon2012-09-28. Gibbings,J.C.(2011).DimensionalAnalysis.Springer.ISBN 978-1-84996-316-9. Originalsources[edit] Vaschy,A.(1892)."Surlesloisdesimilitudeenphysique".AnnalesTélégraphiques.19:25–28. Buckingham,E.(1914)."Onphysicallysimilarsystems;illustrationsoftheuseofdimensionalequations".PhysicalReview.4(4):345–376.Bibcode:1914PhRv....4..345B.doi:10.1103/PhysRev.4.345.hdl:10338.dmlcz/101743. Buckingham,E.(1915)."Theprincipleofsimilitude".Nature.96(2406):396–397.Bibcode:1915Natur..96..396B.doi:10.1038/096396d0.S2CID 3956628. Buckingham,E.(1915)."Modelexperimentsandtheformsofempiricalequations".TransactionsoftheAmericanSocietyofMechanicalEngineers.37:263–296. Taylor,SirG.(1950)."TheFormationofaBlastWavebyaVeryIntenseExplosion.I.TheoreticalDiscussion".ProceedingsoftheRoyalSocietyA.201(1065):159–174.Bibcode:1950RSPSA.201..159T.doi:10.1098/rspa.1950.0049.S2CID 54070514. Taylor,SirG.(1950)."TheFormationofaBlastWavebyaVeryIntenseExplosion.II.TheAtomicExplosionof1945".ProceedingsoftheRoyalSocietyA.201(1065):175–186.Bibcode:1950RSPSA.201..175T.doi:10.1098/rspa.1950.0050. Externallinks[edit] Somereviewsandoriginalsourcesonthehistoryofpitheoremandthetheoryofsimilarity(inRussian) Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Buckingham_π_theorem&oldid=1111403732" Categories:DimensionalanalysisPhysicstheoremsHiddencategories:ArticlesneedingadditionalreferencesfromApril2022AllarticlesneedingadditionalreferencesArticleswithshortdescriptionShortdescriptionisdifferentfromWikidataArticlescontainingproofs Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk English Views ReadEditViewhistory More Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Languages БеларускаяCatalàČeštinaDeutschEestiEspañolفارسیFrançaisItalianoNederlandsPolskiPortuguêsРусскийSlovenščinaSuomiУкраїнська中文 Editlinks