ABSTRACT (original book): Dimensional Analysis Across the Landscape of Physics introduces readers to the powerful idea that almost all physical quantities in science and engineering can be described using only five base dimensions: mass, length, time, charge, and temperature, and combinations thereof. Starting with the basics of how this foundational intellectual concept arises, it illustrates the use of dimensional analysis in approaching the solutions to textbook-level problems in physics and adjacent fields, ranging from introductory courses, through the advanced undergraduate curriculum, to advanced Physics electives.
It covers the core curricular topics of classical mechanics, electricity and magnetism, thermal physics, and quantum mechanics. It includes examples of the use of dimensional analysis applied to topics from other related fields such as geosciences, meteorology, engineering, and biophysics to emphasize the utility of such methods across the proverbial landscape of physics. There is also coverage of more specialized topics, such as advanced quantum mechanics, particle physics, field theory, condensed matter physics, and astrophysics and gravitation.
Many worked examples are included, as well as an extensive array of end-of-chapter problems, with a solution manual available to instructors. In addition to covering the standard topics in the undergraduate curriculum, the book explores how dimensional analysis has been used (and continues to be used) in research across all fields of physics, citing examples from the historical literature and from very recent research results. The work includes extensive references to the original papers for further study, as well as useful ancillary material, including a dimensional analysis ‘dictionary’, brief introductions to data-fitting, and connections to metrology. There is an emphasis throughout on the use of modern symbolic programming to streamline the process of the solving systems of linear equations needed for a dimensional analysis approach, with several Mathematica© templates provided for reader use. CITATION (original book): Robinett, P. R. W. (n.d.). Dimensional Analysis Across the Landscape of Physics: Classic Results, Textbook Examples, and Exploration of Research. United Kingdom: OUP Oxford. ISBN: 9780192693266, 0192693263. https://a.co/d/73wUaI8
It covers the core curricular topics of classical mechanics, electricity and magnetism, thermal physics, and quantum mechanics. It includes examples of the use of dimensional analysis applied to topics from other related fields such as geosciences, meteorology, engineering, and biophysics to emphasize the utility of such methods across the proverbial landscape of physics. There is also coverage of more specialized topics, such as advanced quantum mechanics, particle physics, field theory, condensed matter physics, and astrophysics and gravitation.
Many worked examples are included, as well as an extensive array of end-of-chapter problems, with a solution manual available to instructors. In addition to covering the standard topics in the undergraduate curriculum, the book explores how dimensional analysis has been used (and continues to be used) in research across all fields of physics, citing examples from the historical literature and from very recent research results. The work includes extensive references to the original papers for further study, as well as useful ancillary material, including a dimensional analysis ‘dictionary’, brief introductions to data-fitting, and connections to metrology. There is an emphasis throughout on the use of modern symbolic programming to streamline the process of the solving systems of linear equations needed for a dimensional analysis approach, with several Mathematica© templates provided for reader use. CITATION (original book): Robinett, P. R. W. (n.d.). Dimensional Analysis Across the Landscape of Physics: Classic Results, Textbook Examples, and Exploration of Research. United Kingdom: OUP Oxford. ISBN: 9780192693266, 0192693263. https://a.co/d/73wUaI8
The fact that (almost) all physical quantities in science and engineering applications can be ‘spelled’ with an alphabet of just five letters, namely M (for mass), L (length), T (time), Q (electric charge) and Θ (temperature), is every bit as fundamental a ‘grammatical’ fact in nature as is the genetic code which underlies the structure of DNA and life as we know it.
(A,C,G,T)
The use of dimensional analysis (or DA) in physics can range from simple a posteriori checking that the answer to a problem is right (or at least ‘not wrong’) by comparing physical dimensions on both sides of an equation, to helping to predict (or at least constrain) the dependencies on the M,L,T,Q,Θ dimensions in an a priori manner. This can be done for both classical textbook-level problems, but also when exploring newer research-based examples[1, 2, 3].
As a simple example of the predictive power of DA, consider trying to determine the dependence of the period () of a classical mass-and-spring system (harmonic oscillator) on the mass (), spring constant (), and initial stretch of the system . Assuming that the period can depend on powers of the three dimensional physical parameters in question, and a dimensionless constant , we can write
As a simple example of the predictive power of DA, consider trying to determine the dependence of the period (
τ
m
k
()
x
0
()
C
τ
τ
0
C
τ
α
m
β
k
γ
x
0
α
M
β
M
2
T
γ
L
(
1
)where we display the dimensions of both sides of Eqn. (1). For consistency, the dimensions of each side must match, giving a system of linear equations, namely
M: 0α+β, L: 0 -γ, T: 1 -2β ,
(
2
)which is easily solved (almost by inspection) to yield and . This implies that = and elementary mechanics gives =2π.
α=-β=1/2
γ=0
τ
0
C
τ
m/k
C
τ
While this example is almost trivial to solve, more complex problems, ones involving more variables, and/or more dimensional constraints (from Q and Θ) can benefit from an automated approach to both setting up and solving the linear algebra problem involved in finding the exponent variables. If nothing else, a template program to facilitate repeated use of such DA methods can help mitigate the tedium of having to solve multiple linear algebra problems, all of a very similar format. To this end, we approach the general problem in a similar way, namely writing out (conceptually at least) the M, L, T, Q, Θ dimensions of the desired quantity on the left-hand-side (LHS), as well as those of the constituent variables on the right-hand-side (RHS), ××⋯. In the spirit of the famous Buckingham Pi theorem[4], we note that the ratio of the RHS to the LHS must be dimensionless (dimensional homogeneity, as it’s sometimes called), so that the exponents of the individual M, L, T, Q, Θ terms in the ratio = RHS/LHS must vanish. Enforcing those constraints allows us to set up the corresponding system of linear equations, to which we can apply the appropriate Solve[ ] command to determine (or constrain) the exponents . We illustrate the case of the harmonic oscillatorconsidered above as follows:
α
var
1
β
var
2
γ
var
3
α,β,γ,…
HARMONIC OSCILLATOR - Solve for the dependence of the period of the harmonic oscillator on the mass, spring constant, and initial stretch of the spring
Clear[alpha,beta,gamma,delta,expM,expL,expT,RHS,LHS];(*Particlemass*)mass=M;(*Springconstant*)kspring=M/T^2;(*Initialstretch*)x0=L;(*TheRHSintermsofthesevariables*)RHS=mass^(alpha)*kspring^(beta)*x0^(gamma);(*TheLHSistheperiod,tau*)tau=T;LHS=tau;(*TaketheRHS/LHSratioandinsistthatalldimensionscancel*)ratio=RHS/LHS;expM=Exponent[ratio,M];expL=Exponent[ratio,L];expT=Exponent[ratio,T];Solve[{0expM,0expL,0==expT},{alpha,beta,gamma}]
Out[]=
alpha,beta-,gamma0
1
2
1
2
which is indeed the solution noted above.
We note that we have (for the most part) followed the notation of the Mathematica programs utilized in Ref.[1], using very simple coding structures, which are easily modified and extended to other problems. All that is then required for a given DA problem is the dimensional spelling of any/all variables, and Ref.[1] also includes an extensive ‘dictionary’ giving the M, L, T, Q, Θ content of many physical quantities.
Such methods can be applied not only to canonical textbook level problems, but also to research level questions. For example, no matter what approach is being used to explore relativistic quantum gravity, we know that it must include the fundamental constants associated with each of those three descriptive terms, namely the speed of light () from special relativity, Planck’s constant of quantum mechanics (), and Newton’s constant of gravity (). We can then determine, for example, what combination gives the so-called Planck length.
c
ℏ
G
α
ℏ
β
c
γ
G
PLANCK LENGTH - We solve for the dimensional dependence of the Planck length
Clear[alpha,beta,gamma,expM,expL,expT,RHS,LHS];(*ThisisNewton'sgravitationalconstant*)GNewton=L^3/(M*T^2);(*ThisisPlancks(reduced)constant*)hbar=(M*L^2)/T;(*Thisisthespeedoflight*)clight=L/T;(*ThePlancklengthontheRHSintermsofthesevariables*)RHS=hbar^(alpha)*GNewton^(beta)*clight^(gamma);(*FortheLHSwehave*)LPlanck=L;LHS=LPlanck;(*TaketheRHS/LHSratioandinsistthatalldimensionscancel*)ratio=RHS/LHS;expM=Exponent[ratio,M];expL=Exponent[ratio,L];expT=Exponent[ratio,T];Solve[{0expM,0expL,0expT},{alpha,beta,gamma}]
Out[]=
alpha,beta,gamma-
1
2
1
2
3
2
or ==m. This is an incredibly small length scale which immediately justifies how experimentally challenging it is to probe such effects in any laboratory setting. One of the benefits of having such template programs is that it’s easy to change the quantity considered on the LHS (and its dimensions) to find other Planckian quantities (energy, time, power, etc.).
L
P
1/2
ℏ
1/2
G
-3/2
c
ℏG/
≈1.6×3
c
-35
10
A more experimentally accessible research example involves the “Quantum states of neutrons in the Earth’s gravitational field”[6] where the relevant quantities are the neutron mass (), Planck’s constant (ℏ), but now the gravitational acceleration at the Earth’s surface (). The typical length scale for such bound neutrons, , obtained from the variables in this problem is given as follows:
m
n
g
l
n
GRAVITATIONALLY FOUND NEUTRONS - LENGTH SCALE - Finding the natural length scale for a neutron gravitationally bound near the Earth’ s surface due to quantum mechanics
Clear[alpha,beta,gamma,expM,expL,expT,RHS,LHS];(*GravitationalaccelerationatEarth'ssurface*)gEarth=L/T^2;(*Massofneutron*)mneutron=M;(*RHScontainsthefollowingparameters*)RHS=hbar^(alpha)*gEarth^(beta)*mneutron^(gamma);(*LHShasthecharacteristicboundstatelengthscale*)lneutron=L;LHS=lneutron;(*TaketheRHS/LHSratioandinsistthatalldimensionscancel*)ratio=RHS/LHS;expM=Exponent[ratio,M];expL=Exponent[ratio,L];expT=Exponent[ratio,T];Solve[{0expM,0expL,0expT},{alpha,beta,gamma}]
Out[]=
alpha,beta-,gamma-
2
3
1
3
2
3
or ∼=≈7×m, or of order . This is a length scale which is accessible in the lab and agrees with what’s seen in experiment[6].
l
n
2/3
ℏ
-1/3
g
-2/3
m
n
1/3
g
2
ℏ
2
m
n
-6
10
10μm
It’s not always the case where the number of variables equals the number of constraint equations. One famous example from classical hydrodynamics is the problem of a sphere (radius ) moving with constant speed () through a fluid medium of known density () and viscosity (). The force () required to maintain this motion might depend on these four parameters, but with only three constraints (on the M, L, T dimensions), we don’t expect a unique answer. One can find the general solution in terms of any single exponent, such as:
D
v
ρ
η
F
REYNOLDS NUMBER - Force on a sphere of diameter D moving with speed v in a medium of density and viscosity rho and eta
Clear[alpha,beta,gamma,delta,expM,expL,expT,RHS,LHS];(*Particleradius*)Dsphere=L;(*Particlespeed*)vel=L/T;(*Materialpropertiesoffluid*)viscosity=M/(L*T);density=M/L^3;(*RHScombination*)RHS=density^(alpha)*viscosity^(beta)*Dsphere^(gamma)*vel^(delta);(*LHSisaforce*)force=(M*L)/T^2;LHS=force;(*TaketheRHS/LHSratioandinsistthatalldimensionscancel*)ratio=RHS/LHS;expM=Exponent[ratio,M];expL=Exponent[ratio,L];expT=Exponent[ratio,T];(*Solveintwodifferentways,intermsofalpha,orintermsofbeta*)Solve[{0expM,0expL,0expT},{alpha,beta,gamma,delta}]Solve[{0expM,0expL,0expT},{beta,alpha,gamma,delta}]
These two solutions correspond to
SCHWINGER EFFECT - Find the dimensions of the critial electric field to produce electron/positron pairs from the vacuum
LARMOR RADIATION FORMULA - Calculate the power emitted by a charge undergoing uniform ACCELERATION
This result is originally due to Larmor[9], who also noted that “...In motion with uniform velocity there is no loss...”, meaning no power dissipated from a particle moving at constant speed. We can confirm this statement, in a DA approach, by trivially morphing the code above. In lieu of the uniform acceleration, we simply replace acc with a vel term with dimensions of speed, namely [vel]=L/T. The result is
LARMOR RADIATION FORMULA - CONSTANT SPEED VARIATION - Calculate the power radiated by a charge q moving with constant VELOCITY/SPEED v
where the empty set output (namely { }) means that there is no solution, as expected.
LORENZ CONSTANT FOR THERMAL/ELECTRICAL TRANSPORT MEASUREMENTS - How does the Lorenz constant, the ratio of thermal to electrical conductivities, divided by temperature, depend on fundamental constants like the the electron mass and charge, Boltzmann’ s constant, and the carrier density and scattering time
We have shown examples where template Mathematica programs can be easily modified to explore how dimensional analysis methods can be used to predict (or at least constrain) textbook to research level results in Physics. We have illustrated examples ranging from correct predictions of the complete dimensional dependence on the variables in the problem, to ones identifying important dimensionless ratios, and even finding that no solution is possible in some cases.
References
References
[1] R. W. Robinett, Dimensional Analysis Across the Landscape of Physics: Classic Results, Textbook Examples, and Exploration of Research (Oxford University Press, 2024).
[2] R. W. Robinett, “Dimensional analysis as the other language of physics”, Am. J. Phys. 83 353-361 (2015)
[3] R. W. Robinett, “Quantum fields from dimensional analysis: Applications in condensed matter systems”, Am. J. Phys. 93 to appear (2025).
[4] E. Buckingham, “On physically similar systems: Illustrations of the use of dimensional equations”, Phys. Rev. 4(4), 345-376 (1914).
[5] M. Planck, Ueber irreversible Strahlungsvorgange (On irreversible radiation processes), Ann. Phys. (Berlin) 306 69-122 (1899).
[6] V. V. Nesvizhevsky V V et al., “Quantum states of neutrons in the Earth’s gravitational \feld”, Nature 415 297-299 (2002).
[7] J. Schwinger, “On gauge invariance and vacuum polarization”, Phys. Rev. 82(5) 664-679 (1951).
[8] D. Griffiths, Introduction to Electrodynamics Third Edition (Prentice-Hall, New Jersey, 1999).
[9] J. Larmor, “LXIII On the theory of the magnetic influence on spectra; and on the radiation from moving ions”, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 44 503-512 (1897) .
[10] N. W. Ashcroft and N. D. Mermin, Solid State Physics (Brooks/Cole, Australia, 1976) .
[11] C. Kittel, Introduction to Solid State Physics Fourth edition (Wiley and Sons, New York, 1971).
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Dimensional analysis across the landscape of physics: results, examples, and exploration of research
by Richard Robinett
Wolfram Community, STAFF PICKS, February 26, 2025
https://community.wolfram.com/groups/-/m/t/3400679
by Richard Robinett
Wolfram Community, STAFF PICKS, February 26, 2025
https://community.wolfram.com/groups/-/m/t/3400679