Dimensional analysis with constraints

Dimensional analysis with constrain ts Ump ei Miy amoto Researc h and Education Cen ter for Comprehensiv e Science, Akita Prefectural Univ ersit y , Akita 015-0055, Japan ump ei@akita-pu.ac.jp Abstract W e dev elop a linear-algebraic framew ork for dimensional analysis in systems with constrain ts, particularly when v ariables are numerous or related by implicit relations so that direct elimination is impractical. By expressing b oth dimensional relations and constraints in logarithmic v ariables, the problem is reduced to a linear structure. This formulation yields a simple count of indep enden t dimensionless quan tities and, more imp ortantly , a purely algebraic pro cedure to eliminate redundant ones without trial and error. The method is especially effectiv e for systems with implicit or m ultiple constraints, and is illustrated with the classical drag force problem. 1 In tro duction Dimensional analysis provides a p o w erful to ol for reducing the num b er of v ariables in ph ysical problems without requiring detailed knowledge of gov erning equations. Its cen tral result, the Buc kingham π -theorem, states that a relation among n dimensional v ariables can b e rewritten in terms of n − rank A dimensionless quantities, where A is the dimension matrix of the system [1]. This reduction often rev eals the essential structure of physical la ws and pla ys a k ey role in mo deling, experiment design, and scaling analysis across man y areas of ph ysics and engineering. Despite its generalit y , the π -theorem has an intrinsic limitation: it do es not provide a unique or systematic w a y to select dimensionless quan tities, and it do es not directly incorp orate additional relations among v ariables. In practice, the construction of π -groups t ypically relies on heuristic c hoices such as rep eating v ariables, and differen t selections may lead to equiv alen t but not necessarily minimal or transparen t represen tations. Suc h situations frequen tly arise when v ariables are connected by constitutive relations, definitions, or implicit equations. When the num b er of v ariables is large or the constrain ts are complicated, it may be impractical to eliminate dep enden t v ariables explicitly . In suc h cases, one is naturally led to retain all v ariables and p ostp one the elimination of dep endencies. Instead, one constructs dimensionless quantities at the lev el of all v ariables and then imp oses the constrain ts afterw ard. This pro cedure typically generates a set of candidate dimensionless quan tities that satisfy dimensional inv ariance but are not all indep enden t due to the constraints. Iden tifying and removing redundant dimensionless quan tities then b ecomes a non trivial task. The use of logarithmic v ariables to con v ert m ultiplicativ e structures into linear ones is standard in dimensional analysis and scaling theory , and is closely related to viewing dimensional transformations as linear op erations on exp onen t vectors [2]. In this form u- lation, the dimension matrix and the construction of dimensionless quantities naturally lead to a linear-algebraic structure. How ever, existing approaches do not systematically 1 incorp orate constrain ts into this framew ork in a wa y that directly yields independent dimensionless quan tities. Extensions of dimensional analysis b ey ond its classical form ha v e also b een discussed, including algorithmic approaches for iden tifying dimensionless v ariables [3]. These w orks highligh t b oth the flexibilit y of dimensional analysis and the non-uniqueness of dimen- sionless representations. Nev ertheless, a simple and constructiv e metho d for handling constrain ts and extracting indep enden t dimensionless quan tities remains lac king. In this pap er, w e present a linear-algebraic framew ork for dimensional analysis in con- strained systems. By expressing b oth dimensional relations and constraints in logarithmic v ariables, w e reduce the problem to the study of in tersections of linear subspaces. This leads to a simple expression for the effectiv e num b er of indep enden t dimensionless quan- tities and, more imp ortan tly , to a mec hanical pro cedure for extracting an indep enden t set by eliminating redundan t ones via elemen tary matrix op erations. The nov elty of the presen t work lies in the systematic incorp oration of constraints in to this linear structure and in the resulting constructiv e extraction of indep enden t dimensionless quan tities. The remainder of this pap er is organized as follows. In Sec. 2, w e review the Buck- ingham π -theorem and introduce the logarithmic formulation. In Sec. 3, we extend the framew ork to constrained systems and derive expressions for the effectiv e num b er of di- mensionless quantities. W e then presen t a mechanical pro cedure for eliminating redun- dan t quan tities. In Sec. 4, w e illustrate the metho d with the classical drag force problem. Finally , Sec. 5 summarizes the results and discusses p ossible extensions. 2 Review of Buc kingham π -theorem 2.1 Dimension matrix and logarithmic v ariables W e assume that there are m formal dimensions D 1 , D 2 , . . . , D m and n physical quantities x 1 , x 2 , . . . , x n in the system. The dimension of each physical quantit y is written as J x j K = m Y i =1 D a ij i (1 ≤ j ≤ n ) . (1) The matrix defined b y A : = [ a ij ] ∈ R m × n is called the dimension matrix . Hereafter, w e in tro duce a vector x = ( x 1 , x 2 , . . . , x n ) ∈ R n > 0 , assuming that its comp o- nen ts are p ositive for later conv enience. This assumption is t ypically satisfied in applica- tions where the ph ysical quantities are p ositive. Ho wev er, if a comp onen t can cross zero, the logarithmic form ulation b ecomes problematic and requires a separate treatmen t. Let us define a monomial of ph ysical quan tities written as x v : = n Y j =1 x v j j , (2) where v : = ( v 1 , v 2 , . . . , v n ) ∈ R n is a v ector of exponents. Then, its logarithm is writ- ten as an inner pro duct of the exp onent v ector and logarithmic v ariable y : = ln x : = (ln x 1 , ln x 2 , . . . , ln x n ) ∈ R n : ln x v = n X j =1 v j ln x j = ⟨ v , y ⟩ , (3) 2 where ⟨ , ⟩ denotes the standard Euclidean inner pro duct. F rom the bilinearity of the inner pro duct, the c hange in logarithmic v ariable y → y + δ y results in a change in ln x v as δ ln x v = ⟨ v , δ y ⟩ . (4) 2.2 Dimensionless quan tities and statements of the π -theorem No w, let us consider scaling of dimensions caused b y c hange of units, D i → e λ i D i , λ i ∈ R (1 ≤ i ≤ m ) . (5) F rom the dimensional dep endence of x j on D i , namely Eq. (1), such a c hange of units results in a translation in the logarithmic v ariables: y → y + A ⊤ λ , (6) where ⊤ denotes the transp ose and λ : = ( λ 1 , λ 2 , . . . , λ m ) . W e no w deriv e the condition under whic h the monomial x e is dimensionless. Noting that a dimensionless quantit y is one whose v alue is inv ariant under arbitrary c hanges of units, w e imp ose that x e do es not c hange under y → y + A ⊤ λ : δ ln x e = ⟨ e , A ⊤ λ ⟩ = ⟨ A e , λ ⟩ = 0 . (7) This holds for arbitrary λ if and only if e ∈ ker A . Therefore, the n um b er d of dimension- less quantities in the system is equal to the dimension of ker A . Using the rank-n ullit y theorem, it is obtained as d : = dim ker A = n − rank A. (8) Let { e 1 , e 2 , . . . , e d } b e a basis of ker A , then the Buc kingham π -theorem also states that an y ph ysical relation in the system tak es the form of F ( π 1 , π 2 , . . . , π d ) = 0 , π k : = x e k (1 ≤ k ≤ d ) , (9) where F : R d → R . 2.3 Orthogonal decomp osition of space of logarithmic v ariables Before generalizing to constrained systems, we introduce a useful decomposition of the space of logarithmic v ariables. F rom Eq. (7), any dimensionless quan tit y do es not change when the change of y tak es the form of δ y = A ⊤ λ , namely δ y ∈ im A ⊤ . W e refer to im A ⊤ as the sc aling dir e ction , and decomp ose the space of δ y into this scaling direction and one perp endicular to it: R n = im A ⊤ ⊕ k er A. (10) Note that (im A ⊤ ) ⊥ = ker A from the fundamen tal theorem of linear algebra [4]. W e call the second part in Eq. (10) the dimensionless dir e ction , since motion along this subspace c hanges dimensionless com binations and do es not correspond to scaling transformations. Let us define the matrix E : = [ e 1 e 2 . . . e d ] ∈ R n × d , (11) whose columns constitute a basis of k er A . Then, an arbitrary c hange of log v ariable can b e written as δ y = A ⊤ λ + E η ( λ ∈ R m , η ∈ R d ) . (12) This decomp osition will pla y a cen tral role in what follo ws. 3 3 Generalization to systems with constrain ts 3.1 Constrain t manifold and its scale inv ariance Using a map ϕ : R n → R ℓ , the constrain ts are assumed to take the form of ϕ ( x ) = 0 ℓ , where 0 ℓ is the ℓ -dimensional zero v ector. F urthermore, w e assume that these constrain ts can b e written equiv alen tly as ψ ( y ) = 0 ℓ , (13) where ψ : R n → R ℓ . The constrain t manifold M is defined as a level set of the ab o v e map: M : = ψ − 1 ( 0 ℓ ) . The Jacobian matrix at a p oin t y ∈ M is defined b y J ( y ) : = D ψ ( y ) : = " ∂ ψ i ∂ y j # 1 ≤ i ≤ ℓ, 1 ≤ j ≤ n ∈ R ℓ × n . (14) Assuming that J = D ψ has lo cally constan t rank on M in a neighborho o d of a p oint y ∈ M , the constan t rank theorem implies that M is a submanifold [5] near y and T y M = k er J ( y ) , dim T y M = n − rank J. (15) Note that the fundamen tal theorem of linear algebra implies another decomp osition of R n : R n = ker J ⊕ im J ⊤ . (16) Finally , let us introduce the notion of sc ale invariant c onstr aint . A constrain t is said to be scale in v arian t if, for any λ ∈ R m , the infinitesimal scaling direction A ⊤ λ is tangent to the constrain t manifold. Namely , it satisfies J A ⊤ λ = 0 ℓ . (17) Th us, the kernel of the Jacobian matrix of a scale inv ariant constraint con tains im A ⊤ . Namely , the follo wing holds for a scale in v arian t constrain t: im A ⊤ ⊆ ker J ( scale in v arian t constrain t ) . (18) 3.2 Effectiv e num b er of free dimensionless quan tities Linearizing the constraint 0 ℓ = ψ ( y + δ y ) − ψ ( y ) with resp ect to δ y , one sees that an infinitesimal change of logarithmic v ariables δ y consistent with the constrain t must satisfy J δ y = 0 ℓ . (19) By Eq. (12), an arbitrary infinitesimal v ariation can be decomp osed as δ y = A ⊤ λ + E η . Since the scaling part A ⊤ λ do es not c hange an y dimensionless quan tit y , the dimensionless degrees of freedom are enco ded only in the comp onen t E η . Therefore, to count admissible dimensionless v ariations, it is sufficien t to imp ose Eq. (19) on δ y = E η , which gives J E η = 0 ℓ . (20) 4 Figure 1: A schematic picture of the space of logarithmic v ariables y = ln x . M = ψ − 1 ( 0 ℓ ) ⊂ R n is the constrain t manifold and T y M = k er J is its tangent space at a point y ∈ M , where J = D ψ is the Jacobian matrix. The am bien t space admits the orthogonal decomp ositions R n = k er A ⊕ im A ⊤ and R n = k er J ⊕ im J ⊤ , where A ∈ R m × n is the dimension matrix of the system. The n um b er d eff of admissible dimensionless quan tities is giv en b y the dimension of ker A ∩ ker J , depicted as the in tersection of T y M and k er A (dimensionless direction). F or scale inv ariant constrain ts, the scaling direction im A ⊤ lies in the tangent space, i.e., im A ⊤ ⊆ ker J . Th us, the admissible dimensionless v ariations are parameterized b y those η ∈ R d satisfy- ing Eq. (20). Accordingly , the effective num b er d eff of admissible dimensionless quantities at y is defined by d eff : = dim ker J E . (21) In what follows, we rewrite this quan tit y in several equiv alent forms useful for geometric in terpretation and computation. By construction, the map E induces a natural identification betw een ker J E and k er A ∩ ker J . k er J E = { η ∈ R d : J E η = 0 ℓ } ∼ = { δ y ∈ R n : J δ y = 0 ℓ , δ y ∈ k er A } = ker A ∩ k er J. (22) In particular, eac h η ∈ ker J E corresponds to δ y = E η ∈ k er A ∩ ker J , and vice v ersa. Therefore, d eff defined b y Eq. (21) is written as d eff = dim(ker A ∩ k er J ) . (23) This gives a clear geometric in terpretation of d eff (see Fig. 1). Namely , d eff is the dimension of the in tersection of the dimensionless direction k er A and the tangen t direction ker J allo w ed b y the constrain t. Equation (23) and the rank-n ullit y theorem yield another expression for d eff : d eff = n − rank " A J # . (24) This expression pro vides the most direct w a y to compute d eff . 5 Finally , using the Grassmann dimension formula together with standard results of linear algebra, d eff in Eq. (23) is rewritten as d eff = dim ker A + dim ker J − dim(ker A + ker J ) = ( n − rank A ) + ( n − rank J ) − [ n − dim(ker A + ker J ) ⊥ ] = n − rank A − rank J + dim(im A ⊤ ∩ im J ⊤ ) . (25) This expression for d eff is easy to compare with the result of the Buc kingham π -theorem in Eq. (8). In the case that the constraint is scale in v arian t, Eq. (18) holds. F or scale inv ariant constrain ts, Eq. (18) together with the fundamental theorem of linear algebra implies that im A ⊤ ⊆ ker J = (im J ⊤ ) ⊥ , namely im A ⊤ ∩ im J ⊤ = { 0 n } . Th us, the last term in Eq. (25) v anishes and w e obtain a simpler expression: d eff = n − rank A − rank J ( scale in v arian t constrain t ) . (26) 3.3 Mec hanical elimination of redundan t dimensionless quan ti- ties The discussion so far giv es the effectiv e num b er d eff of admissible dimensionless quantities. Ho w ev er, in applications one often starts from a basis { e 1 , e 2 , . . . , e d } of k er A and the as- so ciated candidate dimensionless quantities { π 1 , π 2 , . . . , π d } and then wishes to determine systematically whic h of them are redundan t under the constrain ts. T o describ e such a pro cedure, we assume that the constraint is scale inv ariant, so that Eq. (18) holds. W riting J ⊤ = [ j 1 j 2 . . . j ℓ ] , from J A ⊤ = O ℓ × m w e obtain A j k = 0 m (1 ≤ k ≤ ℓ ) . (27) Therefore j k is in ker A and written as a linear combination of { e 1 , e 2 , . . . , e d } . Equiv a- len tly , there exists a matrix C ∈ R ℓ × d suc h that J = C E ⊤ . (28) F rom this, C is uniquely determined by C = J E ( E ⊤ E ) − 1 . (29) F rom π k = x e k , the v ector defined b y ln π : = (ln π 1 , ln π 2 , . . . , ln π d ) is written as ln π = E ⊤ y . (30) Substituting Eq. (28) into the infinitesimal form of constrain t J δ y = 0 ℓ and using Eq. (30), w e obtain C δ ln π = 0 ℓ . (31) Th us, all linear relations among the candidate dimensionless quan tities are enco ded in the matrix C . In particular, the rank of C giv es the num b er of redundant directions among the d candidates, and d eff = d − rank C. (32) Since one can easily sho w rank C = rank J from Eqs. (28) and (29), this is consisten t with previous result (26). This observ ation leads to a mec hanical pro cedure for extracting an indep endent set of admissible dimensionless quan tities: 6 1. Construct the dimension matrix A and c ho ose a basis { e 1 , . . . , e d } of ker A and form the matrix E = [ e 1 e 2 . . . e d ] . 2. Compute the Jacobian matrix J of the constraints and v erify the scale inv ariance condition J A ⊤ = 0 . 3. Compute the matrix C from J = C E ⊤ or equiv alen tly C = J E ( E ⊤ E ) − 1 . 4. Reduce C to row ec helon form. Piv ot v ariables are determined by the free v ariables; the non-pivot columns provide one systematic choice of an indep enden t set of di- mensionless quan tities among { π 1 , π 2 , . . . , π d } . This follows from the fact that row reduction iden tifies a basis of the n ull space of C . In this wa y , redundant dimensionless quan tities can be remov ed without trial and error. The pro cedure dep ends only on elementary linear algebra once the basis of k er A and the Jacobian matrix J are giv en. 4 Demonstration b y the drag force in viscous fluid W e revisit the classic problem of the drag force in a viscous fluid (see e.g. [6]) in order to demonstrate ho w the general framew ork presen ted in the previous section w orks. W e consider the drag force F D acting on a b o dy with a characteristic size L and velocity U in a viscous fluid with densit y ρ , viscosity µ and kinematic viscosit y ν . Although there is the defining relation ν : = µ ρ , (33) w e delib erately include ν as an additional v ariable and imp ose Eq. (33) as a constrain t later, in order to illustrate how redundan t v ariables are handled within the present frame- w ork. The fundamental dimensions are taken as M (mass), L (length), T (time), so that m = 3 . Since the dimensions of the ph ysical quan tities are J F D K = ML T − 2 , J ρ K = ML − 3 , J U K = L T − 1 , J L K = L , J µ K = ML − 1 T − 1 , J ν K = L 2 T − 1 , the dimension matrix is A =    1 1 0 0 1 0 1 − 3 1 1 − 1 2 − 2 0 − 1 0 − 1 − 1    . (34) It is straigh tforward to verify that rank A = 3 , and hence d = n − rank A = 3 . Therefore, b efore imp osing constrain ts, there are three candidate dimensionless quantities. A basis of k er A is giv en, for example, b y e 1 = (1 , − 1 , − 2 , − 2 , 0 , 0) , e 2 = (0 , 1 , 1 , 1 , − 1 , 0) , e 3 = (0 , 0 , 1 , 1 , 0 , − 1) , (35) corresp onding to dimensionless quan tities π 1 = F D ρU 2 L 2 , π 2 = ρU L µ , π 3 = U L ν . (36) π 1 and π 2 corresp ond to the drag co efficient C D and the Reynolds num b er R e , resp ectiv ely . 7 The constraint (33) in the logarithmic v ariable y = ln x is written as ψ ( y ) = y 2 − y 5 + y 6 = 0 ( ℓ = 1) . and the Jacobian matrix is J = D ψ ( y ) = h 0 1 0 0 − 1 1 i . (37) whose rank is obviously rank J = 1 . F rom Eqs. (34) and (37), one directly v erifies that J A ⊤ = O 1 × 3 , whic h is the condition for the scale in v ariance of constrain t. Hence, the general form ula (26) is applicable to yield d eff = n − rank A − rank J = 2 . W e no w apply the mec hanical reduction pro cedure. In tro ducing E = [ e 1 e 2 e 3 ] and directly computing C from Eq. (28) or equiv alently Eq. (29), one obtains C = h 0 1 − 1 i . (38) W e see that this C is already in row echelon form and the second column is a pivot column. Th us the constrain t in the C δ ln π = 0 leads to δ ln π 2 − δ ln π 3 = 0 , and so we hav e π 2 π 3 = const . (39) Th us either π 2 or π 3 is redundan t. A conv enient indep endent set is therefore π 1 , π 2 . The ph ysical relation for the drag force can therefore b e written as F ( π 1 , π 2 ) = 0 , where F : R 2 → R , or at least lo cally C D = f ( R e ) , (40) where f : R → R . This is the w ell-known drag law expressed in terms of the drag co efficien t and the Reynolds n um b er. This example clearly illustrates that, although the inclusion of ν increases the n um b er of v ariables to n = 6 and yields d = 3 candidate dimensionless quantities, the constraint ν = µ/ρ reduces the effective num b er to d eff = 2 . The redundancy is detected and remov ed systematically at the level of dimensionless quantities b y the linear-algebraic pro cedure based on the matrix C , without explicitly eliminating v ariables in adv ance. 5 Conclusion W e hav e dev elop ed a linear-algebraic framework for dimensional analysis in constrained systems. The effectiv e n um b er of indep endent dimensionless quan tities is giv en by Eq. (23): d eff = dim(ker A ∩ k er J ) , whic h pro vides a direct geometric c haracterization as the in tersection of the dimensionless direction and the constrain t-compatible directions. W e hav e further sho wn that this quan tity admits simple algebraic expressions that are con v enien t for computation. In particular, for scale inv ariant constraints one obtains d eff = n − rank A − rank J, as in Eq. (26), whic h directly exhibits how constrain ts reduce the degrees of freedom iden tified b y the classical Buckingham π -theorem. T ogether with Eq. (24), these results pro vide a consistent and transparen t wa y to ev aluate the n um b er of indep enden t dimen- sionless quan tities. 8 Bey ond counting, the main con tribution of this w ork is a mec hanical pro cedure for eliminating redundan t dimensionless quantities. By expressing the constrain t in the form J = C E ⊤ as Eq. (28), the dependencies among candidate dimensionless quan tities are enco ded in the matrix C . The relation C δ ln π = 0 ℓ in Eq. (31) then allo ws one to identify redundan t directions, and ro w reduction of C yields an indep enden t subset. This replaces heuristic selection of π -groups b y a systematic and algorithmic metho d based on elemen tary linear algebra. The formulation is particularly adv antageous in situations where the num b er of v ari- ables is large and the constrain ts are implicit or difficult to eliminate explicitly . In suc h cases, direct elimination of v ariables may obscure the underlying structure of the problem or b ecome impractical. In con trast, the present framework allows one to w ork en tirely at the level of dimensionless quan tities and to remov e redundancy in a transparen t and computationally efficien t manner. The key con tribution of the presen t w ork is the systematic incorp oration of constraints in to the linear-algebraic structure of dimensional analysis, together with an explicit and algorithmic pro cedure for extracting indep enden t dimensionless quantities. P ossible extensions include the analysis of situations where the rank of the Jacobian v aries on the constrain t manifold, leading to singular b eha vior, as w ell as applications to more complex systems in which constrain ts arise from constitutive relations or data-driven mo dels. These directions ma y further deep en the understanding of dimensional reduction in constrained settings. A ckno wledgemen ts The author would like to thank Y. Ara y a, K. Hioki, and Y. Shimazaki for useful discussions on related topics. This w ork was supp orted in part by internal researc h funding from Akita Prefectural Univ ersit y . References [1] E. Buckingham. On physically similar systems; illustrations of the use of dimensional equations. Phys. R ev. , 4:345–376, 1914. [2] L. I. Sedo v. Similarity and Dimensional Metho ds in Me chanics . CRC Press, 1993. [3] A. A. Sonin. The Physic al Basis of Dimensional A nalysis . MIT Press, 2001. [4] G. Strang. Intr o duction to Line ar A lgebr a . W ellesley-Cam bridge Press, 5th edition, 2016. [5] J. M. Lee. Intr o duction to Smo oth Manifolds . Springer, 2nd edition, 2013. [6] F. M. White. Fluid Me chanics . McGraw-Hill, 8th edition, 2016. 9