Lie Group Symmetry of Dimensional Analysis

1. Introduction

Dimensional analysis (DA) serves as a foundational analytical tool in physics and engineering, which simplifies complex governing equations and uncovers intrinsic correlations between diverse physical quantities [1,2,3,6,7,11,12,13,14]. Despite its widespread practical application across academia and industry, conventional empirical DA implementations have long neglected the inherent structural features and mathematical essence of DA, leaving its core theoretical dimension insufficiently formalized.

Existing literature has separately advanced the theoretical frameworks of dimensional analysis and scaling Lie group invariance; nevertheless, a systematic and rigorous theoretical integration of these two domains remains absent. This notable research gap motivates the present work, which establishes a rigorous one-to-one correspondence between DA formulations, scaling Lie group invariants, and Lie algebra representation theory. Such theoretical integration represents the core novelty of this study and distinguishes it from all prior investigations. The key limitations of representative existing studies are summarized below.

Bridgman (1922)) [2] formalized DA as an empirical unit transformation framework but did not explore its potential connections to continuous symmetry groups or Lie algebras. This limitation confines DA strictly to a practical operational tool without rigorous geometric and algebraic theoretical underpinnings. Sedov (1959) [3] established the theory of mechanical similarity and linked scaling invariance to macroscopic physical similarity, yet he treated scaling properties as qualitative heuristic principles rather than constructing a complete Lie group structure with formal generators, well-defined invariants, and standardized algebraic rules. In his seminal monograph, Birkhoff (1960) [4] further refined the theoretical system of hydrodynamic similarity and dimensional analysis via systematic discussions of similarity principles and scale invariance in fluid systems. Although his work facilitated DA applications in hydrodynamics, it focused merely on macroscopic similarity phenomena and practical empirical rules, while failing to uncover the latent Lie group symmetry behind dimensional scaling or achieve full algebraic formalization of the DA framework. Barenblatt (1979, 2012) [7,11] developed scaling theories for self-similarity and asymptotic behaviors, with a focus on analytical solutions to scaling-invariant equations. However, his work neither embedded DA into the general Lie group invariant theory framework nor established quantitative mappings between dimensional matrices and Lie algebra representation matrices.

In the field of Lie group analysis, Ovsiannikov (1962) [5] was the firat to study the dimensional analysis via the Lie group analysis without any application. Bluman and Kumei (1989) [8,9] centered exclusively on PDE-oriented symmetry solving and did not regard DA as an independent theoretical framework or explore its intrinsic group-structured properties. Cantwell (2012) [10] extended Lie group methodologies to mechanical and fluid mechanical problems but achieved no substantive theoretical coupling between DA and Lie group theory. Sun (2016) [13] introduced Ovsiannikov’s results [5] in Chinese.

To address this critical research gap, this study explores the intrinsic geometric and algebraic foundations of dimensional analysis and clarifies that the core essence of DA lies in the symmetry simplification of physical laws induced by scaling (stretching) group transformations. Leveraging Lie group and Lie algebra theories, this work constructs a novel theoretical reinterpretation framework for DA. Within this framework, dimensionless quantities are defined as intrinsic absolute invariants that remain unchanged under group operations, and dimensional matrices are correspondingly characterized as representation matrices of the underlying Lie algebra. This study further develops a standardized procedural framework for solving invariant equations using Lie group infinitesimal generators.

Through rigorous theoretical derivation and validation against two canonical physical models—the oscillation period of a simple pendulum and the pressure loss of viscous fluid in pipeline flow—this study demonstrates that the proposed Lie group-based analytical approach can naturally reconstruct and rederive the classical Buckingham $\mathrm{\Pi }$ theorem. More importantly, this approach unveils the essential structural features of orthogonal decoupling among physical quantities, offering a deeper theoretical insight into the fundamental working mechanism of dimensional analysis.

2. Mathematical Foundations of Lie Groups and Infinitesimal Generators

Before exploring the deep structure of dimensional analysis, we first need to establish the mathematical framework of Lie groups and their infinitesimal generators [5,8,9,10,13]. The theoretical core of Lie groups lies in linearizing continuous symmetry transformations through differential operations, thereby utilizing algebraic tools to study geometry and invariants.

2.1. Continuous Transformation Groups and Lie Groups

A one-parameter continuous transformation group defined on a manifold M is a set of the following mappings:

$$x_i^{\prime }={\varphi }_i(x_1,\dots ,x_n;a),i=1,\dots ,n$$

(1)

where a is a continuous parameter. When $a=0$, it corresponds to the identity transformation $x_i^{\prime }=x_i$. If these transformations satisfy the group’s closure, associativity, existence of an identity element and an inverse element, and the mapping is smooth, they constitute a one-parameter Lie group. For the multi-parameter case, the parameters extend to a vector $\mathbf{a}=(a_1,\dots ,a_r)$.

2.2. Infinitesimal Transformations and Generators

The power of Lie groups lies in their local properties. Consider the infinitesimal transformation near the identity element $a=0$, let $a=ϵ$ be an infinitesimal quantity. Expand the transformation at $ϵ=0$ using Taylor series, keeping the first-order term:

$$x_i^{\prime }=x_i+ϵ{\left.{\displaystyle \frac{\partial {\varphi }_i}{\partial a}}\right|}_{a=0}+O\left({ϵ}^2\right)=x_i+ϵ{\xi }_i\left(x\right)+O\left({ϵ}^2\right)$$

(2)

where ${\xi }_i\left(x\right)={\left.{\displaystyle \frac{\partial {\varphi }_i}{\partial a}}\right|}_{a=0}$ is called the coefficient of the infinitesimal transformation.

From this, we can define a first-order partial differential operator:

$$X={\xi }_i\left(x\right){\displaystyle \frac{\partial }{\partial x_i}}$$

(3)

This operator is called the infinitesimal generator of the Lie group. For an r-parameter Lie group, there exist r linearly independent generators $X_1,\dots ,X_r$. The commutators (Lie brackets) between the generators satisfy closure:

$$[X_j,X_k]=X_j{X}_k-X_k{X}_j=C_{jk}^l{X}_l$$

(4)

where $C_{jk}^l$ are the structure constants. The vector space spanned by the generators, together with the Lie bracket operation, constitutes the Lie algebra corresponding to the Lie group.

2.3. Invariants and Invariant Surface Conditions

Under the action of a Lie group, if a function $F(x_1,\dots ,x_n)$ satisfies $F\left(x^{\prime }\right)=F\left(x\right)$, then F is called an invariant of the group. Expanding through infinitesimal transformations, we obtain:

$$F\left(x^{\prime }\right)-F\left(x\right)=ϵXF+O\left({ϵ}^2\right)=0$$

(5)

Ignoring higher-order terms, we obtain the necessary and sufficient condition for F to be an invariant, namely being annihilated by the generator:

$$XF=0$$

(6)

For an r-parameter Lie group, finding independent invariants is equivalent to solving the system of partial differential equations $X_{k}F=0(k=1,\dots ,r)$. According to Lie’s theorem, the number of independent invariants equals the total number of variables n minus the dimension r of the Lie algebra (if the generators are linearly independent).

3. Lie Group Reformulation of Dimensional Analysis

With the mathematical foundations, we can now re-examine dimensional analysis. Dimensional analysis is usually regarded as an engineering technique based on unit conversion or physical intuition, but behind it lies an elegant mathematical structure. From the perspective of Lie groups, dimensional analysis is essentially the study of the invariance of physical laws under the scaling group (stretching group). The Buckingham $\mathrm{\Pi }$ theorem is no longer an empirical rule, but an inevitable result of the invariant theory under Lie group action. The following is a rigorous reformulation of dimensional analysis from the foundations of Lie groups.

3.1. Core Idea: Dimension as Symmetry

In physics, if an equation is a physical law, it must be independent of the choice of units. For example, whether length is measured in meters or feet, Newton’s second law $F=ma$ holds true. The process of changing units is mathematically equivalent to performing scaling transformations on physical quantities. All possible scaling transformations constitute a continuous group—the stretching group. The independence of physical laws from units means that the physical equations maintain formal invariance under the action of this group, i.e., they possess symmetry.

3.2. Lie Group Structure of Scaling Transformations

Suppose a physical system is described by n dimensional physical quantities $x_1,x_2,\dots ,x_n$. Let there be r fundamental dimensions in the system (e.g., $M,L,T$ in mechanics, so $r=3$). We can define an r-parameter scaling Lie group G. The group elements are parameterized by parameters ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_r\in {\mathbb{R}}^{+}$, and their action on the physical quantities is:

$$x_i^{\prime }={\lambda }_1^{a_{i1}}{\lambda }_2^{a_{i2}}\dots {\lambda }_r^{a_{ir}}{x}_i(i=1,2,\dots ,n)$$

(7)

Here, the exponent $a_{ik}$ is exactly the dimensional exponent of the physical quantity $x_i$ in the k-th fundamental dimension. For example, the dimension of velocity v is $L{T}^{-1}$, so $a_{v,L}=1,a_{v,T}=-1$. This constitutes a Lie group representation acting on ${\mathbb{R}}^n$.

3.3. Lie Algebra and Infinitesimal Generators

The properties of a Lie group are determined by its infinitesimal transformations near the identity element, i.e., characterized by its Lie algebra. Let ${\lambda }_k=1+{ϵ}_k$, where ${ϵ}_k$ is an infinitesimal quantity. Taking the logarithm and differentiating with respect to ${ϵ}_k$, we can obtain the infinitesimal generator $X_k$ corresponding to the k-th fundamental dimension:

$$X_k={\displaystyle \sum _{i=1}^n}{a}_{ik}{x}_i{\displaystyle \frac{\partial }{\partial x_i}}(k=1,2,\dots ,r)$$

(8)

These r generators $\{X_1,X_2,\dots ,X_r\}$ span a Lie algebra. Since the scaling operations of different fundamental dimensions commute with each other, i.e., $[X_j,X_k]=0$, this Lie algebra is an Abelian Lie algebra. Key correspondence: The dimensional matrix A of physical quantities, with elements $A_{ik}=a_{ik}$, is exactly the representation matrix of the Lie algebra generators under the basis $\left\{x_i{\displaystyle \frac{\partial }{\partial x_i}}\right\}$.

3.4. Invariants and the Derivation of Buckingham $\mathrm{\Pi }$ Theorem

The physical law $f(x_1,\dots ,x_n)=0$ possesses scaling invariance, meaning it is invariant under the action of the Lie group G. According to Lie theory, the function f is an invariant of the group G if and only if it is annihilated by all generators of the Lie algebra:

$$X_{k}f=0(k=1,\dots ,r)$$

(9)

We need to find the independent solutions of this system of partial differential equations, i.e., to find the absolute invariants (dimensionless numbers $\mathrm{\Pi }$).

Let $\mathrm{\Pi }=x_1^{{\alpha }_1}{x}_2^{{\alpha }_2}\dots x_n^{{\alpha }_n}$, for $\mathrm{\Pi }$ to be an invariant, it must satisfy $X_k\mathrm{\Pi }=0$. Substituting the generator expression:

$$X_k\mathrm{\Pi }=\left({\displaystyle \sum _{i=1}^n}{a}_{ik}{x}_i{\displaystyle \frac{\partial }{\partial x_i}}\right)\mathrm{\Pi }=\left({\displaystyle \sum _{i=1}^n}{a}_{ik}{\alpha }_i\right)\mathrm{\Pi }=0$$

(10)

Since $\mathrm{\Pi }\ne 0$, we obtain the system of linear equations:

$${\displaystyle \sum _{i=1}^n}{a}_{ik}{\alpha }_i=0(k=1,\dots ,r)$$

(11)

Written in matrix form, this is:

$$A^T\mathbf{\alpha }=\mathbf{0}$$

(12)

where $A^T$ is the transpose of the dimensional matrix, and $\mathbf{\alpha }={({\alpha }_1,\dots ,{\alpha }_n)}^T$ is the exponent vector of the dimensionless number. According to the rank-nullity theorem in linear algebra: the number of independent solutions of the system (i.e., the number of invariants) equals the total number of variables n minus the rank of the coefficient matrix $\mathrm{rank}\left(A\right)$. Let $\mathrm{rank}\left(A\right)=r$, then the number of independent invariants is:

$$k=n-r$$

(13)

These k independent invariants ${\mathrm{\Pi }}_1,{\mathrm{\Pi }}_2,\dots ,{\mathrm{\Pi }}_k$ are exactly the dimensionless $\mathrm{\Pi }$ groups in Buckingham’s $\mathrm{\Pi }$ theorem! The physical law $f(x_1,\dots ,x_n)=0$ can be reduced through invariant decomposition to:

$$F({\mathrm{\Pi }}_1,{\mathrm{\Pi }}_2,\dots ,{\mathrm{\Pi }}_k)=0$$

(14)

4. Profound Generalizations from the Lie Group Perspective

Establishing dimensional analysis on the foundation of Lie groups is not only a mathematical rigorization but also provides powerful capabilities for generalization:

(1) Generalized similarity and complete similarity. Traditional dimensional analysis only considers the stretching group (changing units). However, in complicated systems such as fluid mechanics, generalized similarity exists. For example, the boundary layer equations are not only invariant under the stretching group but may also be invariant under translation and rotation groups. Using the Lie group method, one can systematically find all transformation groups that leave the equations invariant, thereby obtaining more powerful similarity solutions than traditional dimensional analysis (e.g., self-similar solutions).

(2) Partial invariance. Sometimes a physical system does not possess complete scaling invariance but is invariant under certain specific one-parameter subgroups.

(3) Discrete symmetry and parity. Dimensional analysis usually ignores the signs of physical quantities (e.g., vector directions). However, within the Lie group framework, discrete groups can be introduced and combined with the continuous stretching group, thereby explaining why certain dimensionless numbers exhibit specific sign behaviors in physics.

It can be said that dimensional analysis is a special case of Lie group symmetry analysis under the most trivial circumstance of the stretching group (changing units). By understanding its Lie group foundations, we elevate dimensional analysis from a technique of manipulating units to a profound geometric method for finding the symmetry reduction of physical systems. The relation between DA and Lie group is summarized the Table 1.

Table 1. Traditional Dimensional Analysis and Lie Group Foundations

Traditional Dimensional
Analysis	Lie Group Foundations
Fundamental units	Parameters ${\lambda }_k$ of the scaling Lie group
Dimensional exponents	Coefficients $a_{ik}$ of the Lie algebra generators
Dimensional matrix A	Lie algebra representation matrix
Dimensionless numbers $\mathrm{\Pi }$	Absolute invariants under group action
Buckingham $\mathrm{\Pi }$ theorem	Dimension of the solution space of the Lie algebra generator equation $X_k\mathrm{\Pi }=0$ (Rank-nullity theorem)
Independence of physical laws from units	Invariance (symmetry) of physical laws under the stretching group G

Table 1. Traditional Dimensional Analysis and Lie Group Foundations

Traditional Dimensional
Analysis	Lie Group Foundations
Fundamental units	Parameters ${\lambda }_k$ of the scaling Lie group
Dimensional exponents	Coefficients $a_{ik}$ of the Lie algebra generators
Dimensional matrix A	Lie algebra representation matrix
Dimensionless numbers $\mathrm{\Pi }$	Absolute invariants under group action
Buckingham $\mathrm{\Pi }$ theorem	Dimension of the solution space of the Lie algebra generator equation $X_k\mathrm{\Pi }=0$ (Rank-nullity theorem)
Independence of physical laws from units	Invariance (symmetry) of physical laws under the stretching group G

5. Example 1: Lie Group Solution for the Period of a Simple Pendulum

To concretize the Lie group method for dimensional analysis, we take the classic problem of a simple pendulum’s period as an example, demonstrating step-by-step how to rigorously derive dimensionless numbers starting from the scaling Lie group, and revealing deep structural insights that traditional dimensional analysis cannot directly provide during this process.

5.1. Problem Description

Suppose there is a simple pendulum with length L and bob mass m, oscillating at a small angle under gravitational acceleration g. We want to solve for its period T. The set of relevant physical variables is $x=(T,L,m,g)$, with a total number of variables $n=4$. The fundamental dimensions are taken as mass $\mathcal{M}$, length $\mathcal{L}$, and time $\mathcal{T}$, so the number of dimensions is $r=3$.

5.2. Constructing the Scaling Lie Group

Changing the unit system is equivalent to performing a stretching transformation on the fundamental dimensions. Let the parameters of the stretching group be ${\lambda }_1$ (corresponding to mass $\mathcal{M}$), ${\lambda }_2$ (corresponding to length $\mathcal{L}$), and ${\lambda }_3$ (corresponding to time $\mathcal{T}$), with ${\lambda }_i\in {\mathbb{R}}^{+}$. The transformations of each physical quantity under the action of the group element $({\lambda }_1,{\lambda }_2,{\lambda }_3)$ are:

• The dimension of period T is $\mathcal{T}$: $T^{\prime }={\lambda }_3^{1}T$
• The dimension of pendulum length L is $\mathcal{L}$: $L^{\prime }={\lambda }_2^{1}L$
• The dimension of mass m is $\mathcal{M}$: $m^{\prime }={\lambda }_1^{1}m$
• The dimension of gravitational acceleration g is $\mathcal{L}{\mathcal{T}}^{-2}$: $g^{\prime }={\lambda }_2^1{\lambda }_3^{-2}g$

This constitutes a 3-parameter Abelian Lie group acting on the physical space ${\mathbb{R}}^4$.

5.3. Deriving the Lie Algebra and Infinitesimal Generators

The local properties of the Lie group are determined by its infinitesimal transformations near the identity element ${\lambda }_i=1$. Let ${\lambda }_i=1+{ϵ}_i$, ignoring higher-order infinitesimals, the transformations become:

$$\begin{array}{cc}\hfill T^{\prime }& =T+{ϵ}_{3}T,L^{\prime }=L+{ϵ}_{2}L,\hfill \\ \hfill m^{\prime }& =m+{ϵ}_{1}m,g^{\prime }=g+({ϵ}_2-2{ϵ}_3)g.\hfill \end{array}$$

(15)

The Lie algebra generators corresponding to the parameters ${ϵ}_1,{ϵ}_2,{ϵ}_3$, $X_k={\sum }_{i=1}^n{a}_{ik}{x}_i{\displaystyle \frac{\partial }{\partial x_i}}$, can be written directly:

1. Mass generator $X_1$ (corresponding to ${ϵ}_1$, only m changes): $X_1=m{\displaystyle \frac{\partial }{\partial m}}$.

2. Length generator $X_2$ (corresponding to ${ϵ}_2$, L and g change): $X_2=L{\displaystyle \frac{\partial }{\partial L}}+g{\displaystyle \frac{\partial }{\partial g}}$.

3. Time generator $X_3$ (corresponding to ${ϵ}_3$, T and g change): $X_3=T{\displaystyle \frac{\partial }{\partial T}}-2g{\displaystyle \frac{\partial }{\partial g}}$.

These generators span the Lie algebra of the stretching group, and they commute with each other: $[X_i,X_j]=0$.

5.4. Solving the Invariant Equations

The physical law $f(T,L,m,g)=0$ is invariant under unit transformations, meaning f is an invariant under the Lie group action. According to Lie theory, the necessary and sufficient condition for f to be an invariant is that it is annihilated by all generators of the Lie algebra:

$$X_{1}f=0,X_{2}f=0,X_{3}f=0$$

(16)

We need to find the independent solutions of these partial differential equations, i.e., the absolute invariants $\mathrm{\Pi }$. Let $\mathrm{\Pi }=T^{{\alpha }_1}{L}^{{\alpha }_2}{m}^{{\alpha }_3}{g}^{{\alpha }_4}$. Substitute $\mathrm{\Pi }$ into the generator action equation $X_k\mathrm{\Pi }=0$:

1. Substituting into $X_1\mathrm{\Pi }=0$, we have $\left(m{\displaystyle \frac{\partial }{\partial m}}\right)T^{{\alpha }_1}{L}^{{\alpha }_2}{m}^{{\alpha }_3}{g}^{{\alpha }_4}={\alpha }_3\mathrm{\Pi }=0$, implies ${\alpha }_3=0$.

2. Substituting into $X_2\mathrm{\Pi }=0$ (knowing ${\alpha }_3=0$), we have $\left(L{\displaystyle \frac{\partial }{\partial L}}+g{\displaystyle \frac{\partial }{\partial g}}\right)T^{{\alpha }_1}{L}^{{\alpha }_2}{g}^{{\alpha }_4}=({\alpha }_2+{\alpha }_4)\mathrm{\Pi }=0$, gives ${\alpha }_2=-{\alpha }_4$.

3. Substituting into $X_3\mathrm{\Pi }=0$, we have $\left(T{\displaystyle \frac{\partial }{\partial T}}-2g{\displaystyle \frac{\partial }{\partial g}}\right)T^{{\alpha }_1}{L}^{{\alpha }_2}{g}^{{\alpha }_4}=({\alpha }_1-2{\alpha }_4)\mathrm{\Pi }=0$, leads ${\alpha }_1=2{\alpha }_4$.

At this point, 3 out of 4 exponents are constrained, leaving 1 degree of freedom (equal to the number of variables 4 minus the number of independent generators 3). Let ${\alpha }_4=1$, then ${\alpha }_1=2,{\alpha }_2=-1,{\alpha }_3=0$. Thus, we obtain the sole independent dimensionless $\mathrm{\Pi }$ number: ${\mathrm{\Pi }}_1=T^2{L}^{-1}{m}^0{g}^1={\displaystyle \frac{g{T}^2}{L}}$.

5.5. Generator Verification and Geometric Significance

We can directly verify that ${\mathrm{\Pi }}_1={\displaystyle \frac{g{T}^2}{L}}$ is indeed an invariant, i.e., check $X_k{\mathrm{\Pi }}_1=0$ as follows:

$$\begin{array}{cc}\hfill X_1{\mathrm{\Pi }}_1& =m{\displaystyle \frac{\partial {\mathrm{\Pi }}_1}{\partial m}}=0\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill X_2{\mathrm{\Pi }}_1& =(L{\partial }_L+g{\partial }_g){\mathrm{\Pi }}_1=L\left(-{\displaystyle \frac{g{T}^2}{L^2}}\right)+g\left({\displaystyle \frac{T^2}{L}}\right)=0\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill X_3{\mathrm{\Pi }}_1& =(T{\partial }_T-2g{\partial }_g){\mathrm{\Pi }}_1=T\left({\displaystyle \frac{2gT}{L}}\right)-2g\left({\displaystyle \frac{T^2}{L}}\right)=0\hfill \end{array}$$

(19)

Geometric significance: In the 4-dimensional variable space $(T,L,m,g)$, the orbits of the Lie group are 3-dimensional (spanned by the generators as the tangent space). The equation $X_k\mathrm{\Pi }=0$ means that the gradient of $\mathrm{\Pi }$, $\nabla \mathrm{\Pi }$, must be orthogonal to all generator vectors. ${\mathrm{\Pi }}_1$ is precisely the 1-dimensional cross-section (invariant manifold) spanning this 3-dimensional orbit surface.

5.6. Reconstructing the Physical Law

According to Lie group invariant theory, the original physical law $f(T,L,m,g)=0$ can be reduced to an equation between invariants: $F\left({\mathrm{\Pi }}_1\right)=0\Rightarrow {\displaystyle \frac{g{T}^2}{L}}=C$, where C is a constant.

But something seems to be missing here? The period of the simple pendulum should also depend on the initial amplitude ${\theta }_0$! This is because ${\theta }_0$ is inherently a dimensionless number. In the language of Lie groups, ${\theta }_0$ is a trivial invariant of the scaling group (all generators acting on it are 0, because an angle is the ratio of arc length to radius, and unit transformations cancel out). Therefore, the complete reduced equation should be: $F({\mathrm{\Pi }}_1,{\theta }_0)=0\Rightarrow {\displaystyle \frac{g{T}^2}{L}}=\Phi \left({\theta }_0\right)$. That is: $T=\sqrt{{\displaystyle \frac{L}{g}}}\sqrt{\Phi \left({\theta }_0\right)}$. When the small angle approximation is applied, $\Phi \left({\theta }_0\right)\approx 4{\pi }^2$, yielding the well-known $T=2\pi \sqrt{{\displaystyle \frac{L}{g}}}$.

5.7. Deep Insights Provided by the Lie Group Method

Through the above Lie group derivation, we not only obtained the results of the $\mathrm{\Pi }$ theorem but also gained structural insights that traditional dimensional analysis cannot directly provide: Why is the period of a simple pendulum independent of mass m? In traditional physics education, this is usually explained by the fact that gravity provides the acceleration $a=F/m$, and mass is canceled out in kinematics. However, from the Lie group perspective, this is an inevitable consequence of the symmetry structure: the mass generator $X_1=m{\displaystyle \frac{\partial }{\partial m}}$ is a completely decoupled, independent generator; it is included neither in $X_2$ (the length generator) nor in $X_3$ (the time generator). This means that in the scaling Lie group, the scaling of mass is completely orthogonal to the scaling of length and time. Since the invariant $\mathrm{\Pi }$ must satisfy $X_1\mathrm{\Pi }=0$, any invariant composed of $(T,L,g)$ cannot include mass m. Given that the period T must be determined by dimensionless invariants, T can only couple with $L,g$ under scaling transformations, thereby mathematically excluding the influence of mass m. This is the charm of the Lie group foundation of dimensional analysis: it elevates physical intuition into rigorous geometric and algebraic theorems.

6. Example 2: Lie Group Solution for Pressure Drop of Viscous Fluid in a Pipe

To further demonstrate the powerful capability of the Lie group method in handling complex systems with multiple variables and dimensions, we take the classic problem in fluid mechanics—the pressure drop of a viscous fluid in a circular pipe—as an example to show the detailed steps of the Lie group solution. This problem will yield multiple dimensionless numbers and profoundly reveal the algebraic orthogonal relationships among them.

6.1. Problem Description

Consider an incompressible viscous fluid in steady flow through a horizontal circular pipe. Let the pipe diameter be D, the pipe length be L, the fluid density be $\rho$, the dynamic viscosity of the fluid be $\mu$, and the average flow velocity be V. We want to solve for the pressure drop $\Delta p$ between the two ends of the pipe. The set of relevant physical variables is $x=(\Delta p,L,D,\rho ,\mu ,V)$, with a total number of variables $n=6$. The fundamental dimensions are still taken as mass $\mathcal{M}$, length $\mathcal{L}$, and time $\mathcal{T}$, so the number of dimensions is $r=3$.

6.2. Constructing the Scaling Lie Group

Let the parameters of the stretching group be ${\lambda }_1$ (corresponding to mass $\mathcal{M}$), ${\lambda }_2$ (corresponding to length $\mathcal{L}$), and ${\lambda }_3$ (corresponding to time $\mathcal{T}$). The dimensions of each physical quantity and their transformations under the action of the group element $({\lambda }_1,{\lambda }_2,{\lambda }_3)$ are:

• The dimension of pressure drop $\Delta p$ is $\mathcal{M}{\mathcal{L}}^{-1}{\mathcal{T}}^{-2}$: $\Delta p^{\prime }={\lambda }_1^1{\lambda }_2^{-1}{\lambda }_3^{-2}\Delta p$
• The dimension of pipe length L is $\mathcal{L}$: $L^{\prime }={\lambda }_2^{1}L$
• The dimension of pipe diameter D is $\mathcal{L}$: $D^{\prime }={\lambda }_2^{1}D$
• The dimension of density $\rho$ is $\mathcal{M}{\mathcal{L}}^{-3}$: ${\rho }^{\prime }={\lambda }_1^1{\lambda }_2^{-3}\rho$
• The dimension of viscosity $\mu$ is $\mathcal{M}{\mathcal{L}}^{-1}{\mathcal{T}}^{-1}$: ${\mu }^{\prime }={\lambda }_1^1{\lambda }_2^{-1}{\lambda }_3^{-1}\mu$
• The dimension of flow velocity V is $\mathcal{L}{\mathcal{T}}^{-1}$: $V^{\prime }={\lambda }_2^1{\lambda }_3^{-1}V$

This constitutes a 3-parameter Abelian Lie group acting on the physical space ${\mathbb{R}}^6$.

6.3. Deriving the Lie Algebra and Infinitesimal Generators

Let ${\lambda }_i=1+{ϵ}_i$, the Lie algebra generators corresponding to the parameters ${ϵ}_1,{ϵ}_2,{ϵ}_3$ can be written directly:

1. Mass generator $X_1$:

$$X_1=\Delta p{\displaystyle \frac{\partial }{\partial \Delta p}}+\rho {\displaystyle \frac{\partial }{\partial \rho }}+\mu {\displaystyle \frac{\partial }{\partial \mu }}$$

(20)

2. Length generator $X_2$:

$$X_2=-\Delta p{\displaystyle \frac{\partial }{\partial \Delta p}}+L{\displaystyle \frac{\partial }{\partial L}}+D{\displaystyle \frac{\partial }{\partial D}}-3\rho {\displaystyle \frac{\partial }{\partial \rho }}-\mu {\displaystyle \frac{\partial }{\partial \mu }}+V{\displaystyle \frac{\partial }{\partial V}}$$

(21)

3. Time generator $X_3$:

$$X_3=-2\Delta p{\displaystyle \frac{\partial }{\partial \Delta p}}-\mu {\displaystyle \frac{\partial }{\partial \mu }}-V{\displaystyle \frac{\partial }{\partial V}}$$

(22)

6.4. Solving the Invariant Equations

We need to find independent invariants $\mathrm{\Pi }={\left(\Delta p\right)}^{{\alpha }_1}{L}^{{\alpha }_2}{D}^{{\alpha }_3}{\rho }^{{\alpha }_4}{\mu }^{{\alpha }_5}{V}^{{\alpha }_6}$ such that they satisfy $X_1\mathrm{\Pi }=0,X_2\mathrm{\Pi }=0,X_3\mathrm{\Pi }=0$. This yields the following system of linear equations:

1. From $X_1\mathrm{\Pi }=0$: ${\alpha }_1+{\alpha }_4+{\alpha }_5=0.$

2. From $X_2\mathrm{\Pi }=0$: $-{\alpha }_1+{\alpha }_2+{\alpha }_3-3{\alpha }_4-{\alpha }_5+{\alpha }_6=0.$

3. From $X_3\mathrm{\Pi }=0$: $-2{\alpha }_1-{\alpha }_5-{\alpha }_6=0.$

We have 6 variables and 3 independent equations, thus there are $6-3=3$ independent dimensionless numbers. By solving this system of linear equations, we can choose specific free variables to construct invariants with clear physical meanings.

Finding invariant 1 (Aspect ratio): Observing generators $X_1$ and $X_3$, neither contains partial derivatives with respect to L and D. This means L and D are decoupled in mass and time scaling. Let ${\alpha }_1={\alpha }_4={\alpha }_5={\alpha }_6=0,{\alpha }_2=1,{\alpha }_3=-1$, substituting into the equations satisfies all of them. Therefore:

$${\mathrm{\Pi }}_1={\displaystyle \frac{L}{D}}.$$

Finding invariant 2 (Reynolds number): Let ${\alpha }_1=0,{\alpha }_2=0$. From Equation (1), we get ${\alpha }_4=-{\alpha }_5$; from Equation (3), we get ${\alpha }_6=-{\alpha }_5$. Substituting into Equation (2) gives ${\alpha }_3-3(-{\alpha }_5)-{\alpha }_5+(-{\alpha }_5)={\alpha }_3+3{\alpha }_5-{\alpha }_5-{\alpha }_5={\alpha }_3+{\alpha }_5=0$, i.e., ${\alpha }_3=-{\alpha }_5$. Let ${\alpha }_5=-1$, then ${\alpha }_4=1,{\alpha }_6=1,{\alpha }_3=1$. Therefore:

$${\mathrm{\Pi }}_2={\displaystyle \frac{\rho VD}{\mu }}\left(\mathrm{Reynolds}\mathrm{number}Re\right).$$

Finding invariant 3 (Euler number): Let ${\alpha }_2=0,{\alpha }_5=0$. From Equation (1), we get ${\alpha }_4=-{\alpha }_1$; from Equation (3), we get ${\alpha }_6=-2{\alpha }_1$. Substituting into Equation (2) gives $-{\alpha }_1+{\alpha }_3-3(-{\alpha }_1)+(-2{\alpha }_1)=-{\alpha }_1+{\alpha }_3+3{\alpha }_1-2{\alpha }_1={\alpha }_3=0$. Let ${\alpha }_1=1$, then ${\alpha }_4=-1,{\alpha }_6=-2,{\alpha }_3=0$. Therefore:

$${\mathrm{\Pi }}_3={\displaystyle \frac{\Delta p}{\rho V^2}}\left(\mathrm{Euler}\mathrm{number}Eu\right).$$

6.5. Reconstructing the Physical Law and the Darcy Friction Factor

According to Lie group invariant theory, the pressure drop law $f(\Delta p,L,D,\rho ,\mu ,V)=0$ can be reduced to an equation among three invariants: $F\left({\displaystyle \frac{L}{D}},{\displaystyle \frac{\rho VD}{\mu }},{\displaystyle \frac{\Delta p}{\rho V^2}}\right)=0$. Typically, we solve for the Euler number as a function of the other two invariants:

$${\displaystyle \frac{\Delta p}{\rho V^2}}=\Phi \left({\displaystyle \frac{L}{D}},{\displaystyle \frac{\rho VD}{\mu }}\right).$$

In fluid mechanics, for fully developed pipe flow, the pressure drop is directly proportional to the pipe length (which is also a manifestation of local stretching invariance). Therefore, the dependence of the function $\Phi$ on $L/D$ must be linear, allowing $L/D$ to be factored out:

$${\displaystyle \frac{\Delta p}{\rho V^2}}={\displaystyle \frac{L}{D}}·\Psi \left({\displaystyle \frac{\rho VD}{\mu }}\right).$$

By convention, a constant $1/2$ is introduced to use the dynamic pressure head ${\displaystyle \frac{1}{2}}\rho V^2$, and the Darcy friction factor is defined as $f=2\Psi \left(Re\right)$, ultimately yielding the famous Darcy-Weisbach equation:

$$\Delta p=f·{\displaystyle \frac{L}{D}}·{\displaystyle \frac{1}{2}}\rho V^2.$$

6.6. Deep Insights from the Lie Group Perspective: Orthogonal Decoupling

In this example, the Lie algebra generators clearly demonstrate the orthogonal decoupling relationship of physical quantities:

• The generators $X_1$ and $X_3$ do not contain terms for L and D. This means that under mass and time scaling transformations, L and D are completely decoupled; they can only be related through the purely geometric length scaling $X_2$. This explains why the aspect ratio $L/D$ can exist as a purely kinematic invariant that is completely independent of the dynamic parameters ($\rho ,\mu ,V$).
• The traditional dimensional matrix elimination method often only provides an algebraic solution, whereas the generator method, by showing the independent action of each infinitesimal transformation, reveals how different dimensionless numbers (geometric $L/D$, kinematic $Re$, dynamic $Eu$) are orthogonal to each other in the Lie algebra space.

Through these two examples, we see that the Lie group method is not merely about obtaining the final dimensionless expressions; it is more like a scalpel, dissecting the intricate dimensional dependencies in physical systems into mutually orthogonal symmetry orbits, thereby revealing the structure of physical laws at the most fundamental mathematical level.

7. Discussions and Conclusions

This study elevates dimensional analysis (DA) from a purely empirical technique to a rigorous symmetry principle grounded in Lie group and Lie algebra theory, establishing three unprecedented explicit mappings absent from prior literature. Specifically, we prove that dimensionless $\mathrm{\Pi }$ groups are not merely arbitrary algebraic combinations but intrinsic absolute invariants annihilated by all Lie algebra generators of the scaling group; we demonstrate that the dimensional matrix of physical quantities exactly serves as the representation matrix of scaling group generators under the basis $x_i{\displaystyle \frac{\partial }{\partial x_i}}$; and we rigorously derive the Buckingham $\mathrm{\Pi }$ theorem as a direct mathematical outcome of Lie group invariant theory and the linear-algebraic rank-nullity theorem, rather than accepting it as an empirical postulate. This work further reveals that the fundamental essence of DA lies in the symmetry reduction of physical laws under the scaling Lie group. While the Abelian nature of the scaling group renders the relevant mathematical operations equivalent to standard linear algebra, the proposed Lie group framework endows dimensional analysis with profound geometric significance, intuitively uncovering the orthogonal decoupling structure inherent to physical quantities. By virtue of these rigorous theoretical mappings, this research lays a unified theoretical foundation for DA, overcoming the inherent limitations of traditional unit transformation methods and enabling systematic investigations into generalized symmetries—including translational, rotational, and discrete symmetries—as well as the dimensional reduction of complex physical systems.