nondimensionalizes eq, replacing original variables ovars with the variables fvars.


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:
  • GeneratedParametersChow to name generated replacement variables
    GeneratedQuantityMagnitudesKhow to name generated quantity factors
    IncludeQuantities{}additional quantities to include
    UnitSystemAutomaticunit 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".


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Basic Examples  (2)

Remove dimensions from the driven oscillator equation:

Compute the replacement rules for the wave equation that yields a dimensionless form:

Scope  (4)

Solve for the nondimensionalized form of the quantum harmonic oscillator:

Examine the replacement rules for this transformation:

Specify a replacement variable for energy to replace C[1] in the solution:

Nondimensionalize algebraic equations:

Examine the solutions for driven RC and LRC circuits:

Compute all properties of this transformation:

Examine the LRC circuit nondimensionalization:

Remove dimensions from lists of equations at once:

Options  (5)

GeneratedParameters  (1)

Control how introduced parameters are named:

Avoid GeneratedParameters entirely by specifying replacements for more QuantityVariable objects:

GeneratedQuantityMagnitudes  (1)

Adjust the names of introduced constants:

Alternatively, specify additional quantities to include when solving for a dimensionless form:

IncludeQuantities  (1)

Add additional quantities that can be used to solve for a dimensionless form:

UnitSystem  (2)

Simplify Newton's law of gravitation using Planck units:

Compare against the standard method solution:

Explore the available natural unit systems and the values of their units:

Applications  (4)

Nondimensionalize Poisson's equation for gravity:

Examine the dimensionless KleinGordon equation:

Simplify the result further by specifying replacement variables for most of the QuantityVariable objects:

Compute the dimensionless form of physical systems:

Nondimensionalize a one-dimensional Schrödinger's equation using Planck units and Hartree units:

Examine the result to check that it is dimensionless:

Compare to the solution in Hartree atomic units:

Properties & Relations  (1)

DimensionalCombinations derives dimensionless combinations of QuantityVariable objects:

Compare to the dimensionalization rules for an equation for linear motion:

Possible Issues  (3)

Equations must be dimensionally balanced:

Replacement variables must be dimensionless:

Supplied variables should be present within the equation:

Neat Examples  (1)

Transform Coulomb's law and Newton's law of gravitation using different sets of natural units:

Compare how the formulas look under different natural unit systems:

Wolfram Research (2018), NondimensionalizationTransform, Wolfram Language function,


Wolfram Research (2018), NondimensionalizationTransform, Wolfram Language function,


Wolfram Language. 2018. "NondimensionalizationTransform." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2018). NondimensionalizationTransform. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2022_nondimensionalizationtransform, author="Wolfram Research", title="{NondimensionalizationTransform}", year="2018", howpublished="\url{}", note=[Accessed: 19-August-2022 ]}


@online{reference.wolfram_2022_nondimensionalizationtransform, organization={Wolfram Research}, title={NondimensionalizationTransform}, year={2018}, url={}, note=[Accessed: 19-August-2022 ]}