--- title: "NondimensionalizationTransform" language: "en" type: "Symbol" summary: "NondimensionalizationTransform[eq, ovars, fvars] nondimensionalizes eq, replacing original variables ovars with the variables fvars. NondimensionalizationTransform[eq, ovars, fvars, prop] returns a property associated with the nondimensionalization of eq." keywords: - nondimensionalization - dedimensionalization - nondimensionalize - dedimensionalize canonical_url: "https://reference.wolfram.com/language/ref/NondimensionalizationTransform.html" source: "Wolfram Language Documentation" related_guides: - title: "Units & Quantities" link: "https://reference.wolfram.com/language/guide/Units.en.md" related_functions: - title: "QuantityVariable" link: "https://reference.wolfram.com/language/ref/QuantityVariable.en.md" - title: "QuantityVariableDimensions" link: "https://reference.wolfram.com/language/ref/QuantityVariableDimensions.en.md" - title: "DimensionalCombinations" link: "https://reference.wolfram.com/language/ref/DimensionalCombinations.en.md" --- # NondimensionalizationTransform NondimensionalizationTransform[eq, ovars, fvars] nondimensionalizes eq, replacing original variables ovars with the variables fvars. NondimensionalizationTransform[eq, ovars, fvars, prop] returns a property associated with the nondimensionalization of eq. ## Details and Options * ``eq`` is an equation or differential equation constructed of ``Quantity`` and ``QuantityVariable`` objects or a list of such expressions. * ``ovars`` is a list of ``QuantityVariable`` objects present in ``eq``. * ``fvars`` is a list of replacement variables for ``ovar``. * In addition to the nondimensionalized equation, ``NondimensionalizationTransform`` can supply the rules for the "forward" transformation from the dimensional form to the nondimensional form and the "reverse" transformation from the new nondimensional form back to the original equation. * ``NondimensionalizationTransform`` supports the following properties: | | | | ---------------------------------- | --------------------------------------------------------- | | "DimensionalizationMultipliers" | Association of multipliers for the reverse transformation | | "DimensionalizationRules" | list of rules for reversing the transformation | | "NondimensionalizationMultipliers" | Association of multipliers for the transformed variables | | "NondimensionalizationRules" | list of rules for nondimensionalizing the equation | | "ReducedForm" | the nondimensionalized equation | * Alternatively ``"PropertyAssociation"`` can be used to return an ``Association`` of the properties. * ``NondimensionalizationTransform`` returns ``"ReducedForm"`` by default. * The following options can be given: | | | | | ---------------------------- | --------- | ------------------------------------------- | | GeneratedParameters | C | how to name generated replacement variables | | GeneratedQuantityMagnitudes | K | how to name generated quantity factors | | IncludeQuantities | {} | additional quantities to include | | UnitSystem | Automatic | unit system used to generate factors | * The ``GeneratedQuantityMagnitudes`` setting is used in cases where it is not possible to remove all dimensions from a variable using the ``QuantityVariable`` and ``Quantity`` objects within the equation. In those cases, new quantities are added to the solution with the symbol provided by the ``GeneratedQuantityMagnitudes`` option. * The ``GeneratedParameters`` setting is used when there are additional ``QuantityVariable`` objects within the equation not included in ``ovar``. ``NondimensionalizationTransform`` removes all ``QuantityVariable`` objects from an equation. Additional ``QuantityVariable`` objects are replaced by variables as specified by the ``GeneratedParameters`` option. * ``IncludeQuantities`` adds additional ``Quantity`` objects to use in generating dimensionless solutions. * ``UnitSystem`` controls the unit system to use for generating multiplicative factors when removing dimensions from the equation. With ``Automatic``, ``NondimensionalizationTransform`` creates the factors from the units and physical quantities present in the equation or specified by ``IncludeQuantities``. * ``UnitSystem`` can also be set to use a natural units system. These options include ``"DeSitterUnits"``, ``"GaussianNaturalUnits"``, ``"GaussianQuantumChromodynamicsUnits"``, ``"HartreeAtomicUnits"``, ``"LorentzHeavisideNaturalUnits"``, ``"LorentzHeavisideQuantumChromodynamicsUnits"``, ``"PlanckUnits"``, ``"RydbergAtomicUnits"`` and ``"StonyUnits"``. --- ## Examples (20) ### Basic Examples (2) Remove dimensions from the driven oscillator equation: ```wl In[1]:= NondimensionalizationTransform[ω x[t] + a Derivative[1][x][t] == A f[t], {x, t, f}, {y, u, g}] Out[1]= (y[u]/a) + Derivative[1][y][u] == (A g[u]/a) ``` --- Compute the replacement rules for the wave equation that yields a dimensionless form: ```wl In[1]:= waveequation = u^(0, 2)[x, t] == cp^2 u^(2, 0)[x, t]; In[2]:= NondimensionalizationTransform[waveequation, {u, x, t}, {v, y, u}, "NondimensionalizationRules"] Out[2]= {u^(0, 2)[x, t] -> (Quantity[K[1]/K[2]^2, "Pascals"/"Meters"^2]) cp^2 v^(0, 2)[y, u], u^(2, 0)[x, t] -> (Quantity[K[1]/K[2]^2, "Pascals"/"Meters"^2]) v^(2, 0)[y, u], u -> v (Quantity[K[1], "Pascals"]), x -> y (Quantity[K[2], "Meters"]), t -> (u (Quantity[K[2], "Meters"])/cp)} ``` ### Scope (4) Solve for the nondimensionalized form of the quantum harmonic oscillator: ```wl In[1]:= odeHO = (1/2) m x^2 ω^2 ψ[x] + ((Quantity[-(1/2), "ReducedPlanckConstant"^2]) Derivative[2][ψ][x]/m) == E ψ[x]; In[2]:= NondimensionalizationTransform[odeHO, {ψ, x} , {Ψ, ξ}] Out[2]= -ξ^2 Ψ[ξ] + Derivative[2][Ψ][ξ] == -2 C[1] Ψ[ξ] ``` Examine the replacement rules for this transformation: ```wl In[3]:= NondimensionalizationTransform[odeHO, {ψ, x}, {Ψ, ξ}, "NondimensionalizationMultipliers"] Out[3]= <|ψ -> (Quantity[1, 1/Sqrt["ReducedPlanckConstant"]]) m^1 / 4 ((Quantity[1, "ReducedPlanckConstant"])^1 / 4 ω^1 / 4), x -> (Quantity[1, "ReducedPlanckConstant"]/Sqrt[m] (Sqrt[Quantity[1, "ReducedPlanckConstant"]] Sqrt[ω])), E -> (Quantity[1, "ReducedPlanckConstant"]) ω|> ``` Specify a replacement variable for energy to replace ``C[1]`` in the solution: ```wl In[4]:= NondimensionalizationTransform[odeHO, {ψ, x, E}, {Ψ, ξ, e}] Out[4]= -ξ^2 Ψ[ξ] + Derivative[2][Ψ][ξ] == -2 e Ψ[ξ] ``` --- Nondimensionalize algebraic equations: ```wl In[1]:= accelerationSpeedDistance = s == s0 + (1/2) a t^2 + t v; In[2]:= NondimensionalizationTransform[accelerationSpeedDistance, {a, v, t, s0}, {b, w, u, d0}] Out[2]= (1/d0) == 1 + (b u^2/2 d0) + (u w/d0) ``` --- Examine the solutions for driven RC and LRC circuits: ```wl In[1]:= τ = QuantityVariable["t", "Time"]; q = QuantityVariable["Q", "ElectricCharge"]; v = QuantityVariable["V", "ElectricPotential"]; r = QuantityVariable["R", "ElectricResistance"]; c = QuantityVariable["C", "ElectricCapacitance"]; l = QuantityVariable["L", "MagneticInductance"]; In[2]:= odeRC = r q'[τ] + q[τ] / c == v[τ] Out[2]= (Q[t]/C) + R Derivative[1][Q][t] == V[t] In[3]:= odeLRC = l q''[τ] + r q'[τ] + q[τ] / c == v[τ] Out[3]= (Q[t]/C) + R Derivative[1][Q][t] + L Derivative[2][Q][t] == V[t] In[4]:= NondimensionalizationTransform[odeRC, {q, v, τ}, {x, f, u}] Out[4]= x[u] + Derivative[1][x][u] == f[u] ``` Find the properties available: ```wl In[5]:= NondimensionalizationTransform["Properties"] Out[5]= {"DimensionalizationMultipliers", "DimensionalizationRules", "NondimensionalizationMultipliers", "NondimensionalizationRules", "PropertyAssociation", "ReducedForm"} ``` Compute all properties of this transformation: ```wl In[6]:= NondimensionalizationTransform[odeRC, {q, v, τ}, {x, f, u}, "PropertyAssociation"] Out[6]= <|"ReducedForm" -> x[u] + Derivative[1][x][u] == f[u], "NondimensionalizationRules" -> {Derivative[1][Q][t] -> ((Quantity[K[1], "Coulombs"]) Derivative[1][x][u]/C R), Q[t] -> (Quantity[K[1], "Coulombs"]) x[u], V[t] -> (f[u] (Quantity[K[1], "Coulomb ... mensionalizationMultipliers" -> <|Q -> Quantity[K[1], "Coulombs"], V -> (Quantity[K[1], "Coulombs"]/C), t -> C R|>, "DimensionalizationMultipliers" -> <|x -> Quantity[1/K[1], 1/"Coulombs"], f -> (Quantity[1/K[1], 1/"Coulombs"]) C, u -> (1/C R)|>|> ``` Examine the LRC circuit nondimensionalization: ```wl In[7]:= NondimensionalizationTransform[odeLRC, {q, v, τ}, {x, f, u}, "PropertyAssociation"] Out[7]= <|"ReducedForm" -> (x[u]/C[1]) + Derivative[1][x][u] + Derivative[2][x][u] == f[u], "NondimensionalizationRules" -> {Derivative[1][Q][t] -> ((Quantity[K[1], "Coulombs"]) R Derivative[1][x][u]/L), Derivative[2][Q][t] -> ((Quantity[K[1], "Coulombs"]) ... Coulombs"]) R^2/L), t -> Sqrt[L] (Sqrt[L] R^-(2/2)), C -> (L/R^2)|>, "DimensionalizationMultipliers" -> <|x -> Quantity[1/K[1], 1/"Coulombs"], f -> ((Quantity[1/K[1], 1/"Coulombs"]) L/R^2), u -> (R^-(2 (-1)/2)/Sqrt[L] Sqrt[L]), C[1] -> (R^2/L)|>|> ``` --- Remove dimensions from lists of equations at once: ```wl In[1]:= ψ = QuantityVariable["ψ", "Length" ^ (-1 / 2)]; ψaux = QuantityVariable[Subscript["ψ", "aux"], "Length" ^ (-1 / 2) / "Length"]; \[ScriptX] = QuantityVariable["x", "Length"]; \[ScriptM] = QuantityVariable["m", "Mass"]; ω = QuantityVariable["ω", "Frequency"]; ℰ = QuantityVariable["E", "Energy"]; In[2]:= odeHO = {-Quantity[1, "ReducedPlanckConstant"] ^ 2 / (2 \[ScriptM]) ψaux'[\[ScriptX]] + 1 / 2 \[ScriptM] ω ^ 2 \[ScriptX] ^ 2 ψ[\[ScriptX]] == ℰ ψ[\[ScriptX]], ψaux[\[ScriptX]] == ψ'[\[ScriptX]]}; In[3]:= NondimensionalizationTransform[odeHO, {ψ, ψaux, \[ScriptX]}, {Ψ, Ψaux, ξ}] Out[3]= {-ξ^2 Ψ[ξ] + Derivative[1][Ψaux][ξ] == -2 C[1] Ψ[ξ], Ψaux[ξ] == Derivative[1][Ψ][ξ]} ``` ### Options (5) #### GeneratedParameters (1) Control how introduced parameters are named: ```wl In[1]:= eq = (1/2) m x^2 ω^2 ψ[x] + ((Quantity[-(1/2), "ReducedPlanckConstant"^2]) Derivative[2][ψ][x]/m) == E ψ[x]; In[2]:= NondimensionalizationTransform[eq, {ψ, x}, {Ψ, ξ}, GeneratedParameters -> Z] Out[2]= -ξ^2 Ψ[ξ] + Derivative[2][Ψ][ξ] == -2 Z[1] Ψ[ξ] ``` Avoid ``GeneratedParameters`` entirely by specifying replacements for more ``QuantityVariable`` objects: ```wl In[3]:= NondimensionalizationTransform[eq, {ψ, x, E} , {Ψ, ξ, e}, GeneratedParameters -> Z] Out[3]= -ξ^2 Ψ[ξ] + Derivative[2][Ψ][ξ] == -2 e Ψ[ξ] ``` #### GeneratedQuantityMagnitudes (1) Adjust the names of introduced constants: ```wl In[1]:= waveEquation = u^(0, 2)[x, t] == cp^2 u^(2, 0)[x, t]; NondimensionalizationTransform[waveEquation, {u, x, t}, {v, y, u}, "NondimensionalizationMultipliers", GeneratedQuantityMagnitudes -> "foo"] Out[1]= <|u -> Quantity["foo"[1], "Pascals"], x -> Quantity["foo"[2], "Meters"], t -> (Quantity["foo"[2], "Meters"]/cp)|> ``` Alternatively, specify additional quantities to include when solving for a dimensionless form: ```wl In[2]:= NondimensionalizationTransform[waveEquation, {u, x, t}, {v, y, u}, "NondimensionalizationMultipliers", IncludeQuantities -> {Quantity[1, "Atmospheres"], Quantity[1, "PlanckLength"]}] Out[2]= <|u -> Quantity[1, "Atmospheres"], x -> Quantity[1, "PlanckLength"], t -> (Quantity[1, "PlanckLength"]/cp)|> ``` #### IncludeQuantities (1) Add additional quantities that can be used to solve for a dimensionless form: ```wl In[1]:= waveEquation = u^(0, 2)[x, t] == cp^2 u^(2, 0)[x, t]; NondimensionalizationTransform[waveEquation, {u, x, t}, {v, y, u}, IncludeQuantities -> {"Seconds"}] Out[1]= v^(0, 2)[y, u] == v^(2, 0)[y, u] ``` #### UnitSystem (2) Simplify Newton's law of gravitation using Planck units: ```wl In[1]:= F = QuantityVariable["F", "Force"]; m1 = QuantityVariable[Subscript["m", 1], "Mass"]; m2 = QuantityVariable[Subscript["m", 2], "Mass"]; r = QuantityVariable["r", "Length"]; G = Quantity[1, "GravitationalConstant"]; In[2]:= grav = F == G m1 m2 / r ^ 2 Out[2]= F == ((Quantity[1, "GravitationalConstant"]) Subscript[m, 1] Subscript[m, 2]/r^2) In[3]:= NondimensionalizationTransform[grav, {F, m1, m2, r}, {f, mp1, mp2, ρ}, "NondimensionalizationMultipliers", UnitSystem -> "PlanckUnits"] Out[3]= <|F -> Quantity[1, "SpeedOfLight"^4/"GravitationalConstant"], Subscript[m, 1] -> Quantity[1, (Sqrt["ReducedPlanckConstant"]*Sqrt["SpeedOfLight"])/Sqrt["GravitationalConstant"]], Subscript[m, 2] -> Quantity[1, (Sqrt["ReducedPlanckConstant"]*Sqrt["SpeedOfLight"])/Sqrt["GravitationalConstant"]], r -> Quantity[1, (Sqrt["GravitationalConstant"]*Sqrt["ReducedPlanckConstant"])/"SpeedOfLight"^(3/2)]|> ``` Compare against the standard method solution: ```wl In[4]:= NondimensionalizationTransform[grav, {F, m1, m2, r}, {f, mp1, mp2, ρ}, "NondimensionalizationMultipliers"] Out[4]= <|F -> Quantity[K[1], "Newtons"], Subscript[m, 1] -> Quantity[K[2], "Kilograms"], Subscript[m, 2] -> Quantity[K[2], "Kilograms"], r -> Quantity[K[2]/Sqrt[K[1]], (Sqrt["GravitationalConstant"]*"Kilograms")/Sqrt["Newtons"]]|> ``` --- Explore the available natural unit systems and the values of their units: ```wl In[1]:= SlideView[{...}] Out[1]= DynamicModule[«3»] ``` ### Applications (4) Nondimensionalize Poisson's equation for gravity: ```wl In[1]:= x = QuantityVariable["x", "Length"]; y = QuantityVariable["y", "Length"]; z = QuantityVariable["z", "Length"]; g = QuantityVariable["ϕ", "GravitationalPotential"]; ρ = QuantityVariable["ρ", "MassDensity"]; In[2]:= peq = Laplacian[g[x, y, z], {x, y, z}] == 4 Pi Quantity[1, "GravitationalConstant"]ρ Out[2]= ϕ^(0, 0, 2)[x, y, z] + ϕ^(0, 2, 0)[x, y, z] + ϕ^(2, 0, 0)[x, y, z] == (Quantity[4*Pi, "GravitationalConstant"]) ρ In[3]:= NondimensionalizationTransform[peq, {g, ρ}, {γ, r}] Out[3]= γ^(0, 0, 2)[C[1], C[2], C[3]] + (γ^(0, 2, 0)[C[1], C[2], C[3]]/C[3]^2) + (γ^(2, 0, 0)[C[1], C[2], C[3]]/C[2]^2 C[3]^2) == 4 π r ``` --- Examine the dimensionless Klein–Gordon equation: ```wl In[1]:= x = QuantityVariable["x", "Length"]; y = QuantityVariable["y", "Length"]; z = QuantityVariable["z", "Length"]; t = QuantityVariable["t", "Time"]; m = QuantityVariable["m", "Mass"]; ψ = QuantityVariable["ψ", "Length" ^ (-1 / 2)]; In[2]:= eq = 1 / Quantity[1, "SpeedOfLight"] ^ 2D[ψ[x, y, z, t], t, t] - Laplacian[ψ[x, y, z, t], {x, y, z}] + m ^ 2 Quantity[1, "SpeedOfLight"^2] / Quantity[1, "ReducedPlanckConstant"^2]ψ[x, y, z, t] Out[2]= (Quantity[1, "SpeedOfLight"^2/"ReducedPlanckConstant"^2]) m^2 ψ[x, y, z, t] + (Quantity[1, 1/"SpeedOfLight"^2]) ψ^(0, 0, 0, 2)[x, y, z, t] - ψ^(0, 0, 2, 0)[x, y, z, t] - ψ^(0, 2, 0, 0)[x, y, z, t] - ψ^(2, 0, 0, 0)[x, y, z, t] In[3]:= NondimensionalizationTransform[eq, {ψ, t}, {Ψ, τ}] Out[3]= (C[1]^2 Ψ[C[2], C[3], C[4], τ]/C[2]^2 C[3]^2 C[4]^2) + Ψ^(0, 0, 0, 2)[C[2], C[3], C[4], τ] - Ψ^(0, 0, 2, 0)[C[2], C[3], C[4], τ] - (Ψ^(0, 2, 0, 0)[C[2], C[3], C[4], τ]/C[4]^2) - (Ψ^(2, 0, 0, 0)[C[2], C[3], C[4], τ]/C[3]^2 C[4]^2) == 0 ``` Simplify the result further by specifying replacement variables for most of the ``QuantityVariable`` objects: ```wl In[4]:= NondimensionalizationTransform[eq, {ψ, t, x, y, z}, {Ψ, τ, ξ, η, ζ}] Out[4]= Ψ[ξ, η, ζ, τ] + Ψ^(0, 0, 0, 2)[ξ, η, ζ, τ] - Ψ^(0, 0, 2, 0)[ξ, η, ζ, τ] - Ψ^(0, 2, 0, 0)[ξ, η, ζ, τ] - Ψ^(2, 0, 0, 0)[ξ, η, ζ, τ] == 0 ``` --- Compute the dimensionless form of physical systems: ```wl In[1]:= eq = Expand /@ Entity["PhysicalSystem", "VanDerWaalsGas"][EntityProperty["PhysicalSystem", "EquationOfState"]] Out[1]= -b n p - (a b n^3/V^2) + (a n^2/V) + p V == n R T In[2]:= NondimensionalizationTransform[eq, {p, V, T, n}, {\[ScriptP], \[ScriptV], \[ScriptT], ν}] Out[2]= 1 - (\[ScriptV]/ν) - (ν/\[ScriptP] \[ScriptV]) + (ν^2/\[ScriptP] \[ScriptV]^2) == -(\[ScriptT]/\[ScriptP]) ``` --- Nondimensionalize a one-dimensional Schrödinger's equation using Planck units and Hartree units: ```wl In[1]:= r = QuantityVariable["r", "Length"]; t = QuantityVariable["t", "Time"]; me = Quantity[1, "ElectronMass"]; ψ = QuantityVariable["ψ", "Length" ^ (-1 / 2)]; e0 = Quantity[4 Pi, "ElectricConstant"]; e = Quantity[1, "ElementaryCharge"]; In[2]:= eq = I Quantity[1, "ReducedPlanckConstant"]D[ψ[r, t], t] == -Quantity[1, "ReducedPlanckConstant"] ^ 2 / (2 me)D[ψ[r, t], {r, 2}] + e ^ 2 / (e0 r) ψ[r, t] Out[2]= (Quantity[I, "ReducedPlanckConstant"]) ψ^(0, 1)[r, t] == ((Quantity[1/(4*Pi), "ElementaryCharge"^2/"ElectricConstant"]) ψ[r, t]/r) + (Quantity[-(1/2), "ReducedPlanckConstant"^2/"ElectronMass"]) ψ^(2, 0)[r, t] In[3]:= NondimensionalizationTransform[eq, {ψ, r, t}, {Ψ, ρ, τ}, UnitSystem -> "PlanckUnits"] Out[3]= (Quantity[-2*I, ("ElectronMass"*Sqrt["GravitationalConstant"])/ (Sqrt["ReducedPlanckConstant"]*Sqrt["SpeedOfLight"])]) Ψ^(0, 1)[ρ, τ] == ((Quantity[-(1/(2*Pi)), ("ElectronMass"*"ElementaryCharge"^2*Sqrt["GravitationalConstant"])/ ("ElectricConstant"*"ReducedPlanckConstant"^(3/2)*"SpeedOfLight"^(3/2))]) Ψ[ρ, τ]/ρ) + Ψ^(2, 0)[ρ, τ] ``` Examine the result to check that it is dimensionless: ```wl In[4]:= % /. q_Quantity :> UnitConvert[q, "SIBase"] Out[4]= -8.370786403483431`4.33268877506498*^-23 I Ψ^(0, 1)[ρ, τ] == -(6.1084579650684`4.332230091339868*^-25 Ψ[ρ, τ]/ρ) + Ψ^(2, 0)[ρ, τ] ``` Compare to the solution in Hartree atomic units: ```wl In[5]:= NondimensionalizationTransform[eq, {ψ, r, t}, {Ψ, ρ, τ}, UnitSystem -> "HartreeAtomicUnits"] Out[5]= -2 I Ψ^(0, 1)[ρ, τ] == -(2 Ψ[ρ, τ]/ρ) + Ψ^(2, 0)[ρ, τ] ``` ### Properties & Relations (1) ``DimensionalCombinations`` derives dimensionless combinations of ``QuantityVariable`` objects: ```wl In[1]:= s = QuantityVariable["s", "Length"]; s0 = QuantityVariable["s0", "Length"]; a = QuantityVariable["a", "Acceleration"]; t = QuantityVariable["t", "Time"]; v = QuantityVariable["v", "Speed"]; In[2]:= DimensionalCombinations[{s, s0, a, t, v}, GeneratedParameters -> None] Out[2]= {(s0/s), (Sqrt[a] t/Sqrt[s]), (a s/v^2), (s/t v), (Sqrt[a] t/Sqrt[s0]), (a s0/v^2), (s0/t v), (v/a t)} ``` Compare to the dimensionalization rules for an equation for linear motion: ```wl In[3]:= accelerationSpeedDistance = s == s0 + (1/2) a t^2 + t v; In[4]:= NondimensionalizationTransform[accelerationSpeedDistance, {a, t, s0}, {b, u, d0}, "DimensionalizationRules"] Out[4]= {b -> (a s/v^2), u -> (t v/s), d0 -> (s0/s)} ``` ### Possible Issues (3) Equations must be dimensionally balanced: ```wl In[1]:= Q1 = QuantityVariable["Q", "ElectricCharge"]; t1 = QuantityVariable["t", "Time"]; In[2]:= NondimensionalizationTransform[Q1 - t1 == 0, {Q1, t1}, {Q2, t2}] ``` NondimensionalizationTransform::unbal: Equation is dimensionally unbalanced. ```wl Out[2]= NondimensionalizationTransform[Q - t == 0, {Q, t}, {Q2, t2}] ``` --- Replacement variables must be dimensionless: ```wl In[1]:= NondimensionalizationTransform[(d/t) == v, {d, t}, {Quantity[1, "Meters"], t2}] ``` NondimensionalizationTransform::qv: Replacement variables include elements with physical dimensions. ```wl Out[1]= NondimensionalizationTransform[(d/t) == v, {d, t}, {Quantity[1, "Meters"], t2}] ``` --- Supplied variables should be present within the equation: ```wl In[1]:= NondimensionalizationTransform[(d/t) == v, {d, QuantityVariable["Q", "ElectricCharge"]}, {d2, Q2}] ``` NondimensionalizationTransform::invequ: Equation is missing one or more variables {d,Q}. ```wl Out[1]= NondimensionalizationTransform[(d/t) == v, {d, Q}, {d2, Q2}] ``` ### Neat Examples (1) Transform Coulomb's law and Newton's law of gravitation using different sets of natural units: ```wl In[1]:= naturalunits = {"PlanckUnits", "HartreeAtomicUnits", "RydbergAtomicUnits", "LorentzHeavisideNaturalUnits", "GaussianNaturalUnits"}; In[2]:= cleq = First@FormulaData["CoulombsLaw"] Out[2]= F == ((Quantity[1/(4*Pi), 1/"ElectricConstant"]) Subscript[Q, 1] Subscript[Q, 2]/r^2) In[3]:= nlogeq = FormulaData["NewtonsLawOfUniversalGravitation"] Out[3]= F == ((Quantity[1, "GravitationalConstant"]) Subscript[m, 1] Subscript[m, 2]/r^2) In[4]:= coulomb = NondimensionalizationTransform[cleq, {F, Subscript[Q, 1], Subscript[Q, 2], r}, {f, q1, q2, ρ}, UnitSystem -> #]& /@ naturalunits; gravity = NondimensionalizationTransform[nlogeq, {F, Subscript[m, 1], Subscript[m, 2], r}, {f, m1, m2, ρ}, UnitSystem -> #]& /@ naturalunits; ``` Compare how the formulas look under different natural unit systems: ```wl In[5]:= Grid[Prepend[Transpose[{naturalunits, gravity, coulomb}], {"unit system", "gravitational force", "electrostatic force"}], Dividers -> Center] Out[5]= | | | | | ------------------------------ | --------------------------------------------------------------- | ------------------ ... isideNaturalUnits" | 1.4906218399548641`4.031770136065315*^56 f == (m1 m2/ρ^2) | 12.566371 f == (q1 q2/ρ^2) | | "GaussianNaturalUnits" | 1.4906218399548641`4.031770136065315*^56 f == (m1 m2/ρ^2) | 1.000000 f == (q1 q2/ρ^2) | ``` ## See Also * [`QuantityVariable`](https://reference.wolfram.com/language/ref/QuantityVariable.en.md) * [`QuantityVariableDimensions`](https://reference.wolfram.com/language/ref/QuantityVariableDimensions.en.md) * [`DimensionalCombinations`](https://reference.wolfram.com/language/ref/DimensionalCombinations.en.md) ## Related Guides * [Units & Quantities](https://reference.wolfram.com/language/guide/Units.en.md) ## History * [Introduced in 2018 (11.3)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn113.en.md)