DimensionalCombinations

DimensionalCombinations[{pq1,pq2,}]

returns the possible combinations of the list of physical quantities pqi that are dimensionless.

DimensionalCombinations[{pq1,pq2,},dim]

returns the possible combinations of the list of physical quantities pqi that match the dimensions of physical quantity dim.

Details and Options

  • Physical quantities can be valid QuantityVariable objects or physical quantity strings.
  • dim can be a QuantityVariable object. It can also be a combination of QuantityVariable objects or their derivatives.
  • Solutions are determined by the physical quantity components in unit dimensions purely mathematically and have no guarantee of physical significance.
  • Physical dimensions include: "AmountUnit", "AngleUnit", "ElectricCurrentUnit", "InformationUnit", "LengthUnit", "LuminousIntensityUnit", "MassUnit", "MoneyUnit", "SolidAngleUnit", "TemperatureDifferenceUnit", "TemperatureUnit", and "TimeUnit".
  • Dimensionless physical quantities will not be used in the solution.
  • The following options can be given:
  • GeneratedParametersChow to name parameters that are generated
    IncludeQuantities{}additional quantities to include
  • GeneratedParameters takes the option None, which returns a list of parameter-free solutions.
  • IncludeQuantities allows quantity values and constants to be included in the combinations.
  • The setting "PhysicalConstants" for IncludeQuantities includes the quantities Quantity["BoltzmannConstant"], Quantity["ElectricConstant"], Quantity["GravitationalConstant"], Quantity["MagneticConstant"], Quantity["PlanckConstant"], and Quantity["SpeedOfLight"].

Examples

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Basic Examples  (1)

Determine the combination of physical quantities that are dimensionally equivalent to energy:

Find all combinations of physical quantities that result in a dimensionless expression:

Discover if a dimensionless expression is possible with a set of physical quantities:

Scope  (2)

Use any combination of QuantityVariable objects or physical quantity strings:

The target physical dimensions can be specified as a combination of physical quantities:

Derivative objects may also be included in the expression:

Options  (5)

GeneratedParameters  (3)

Use a different symbol for parameters:

By default, a generic solution is returned:

Use GeneratedParameters->None to get specific solutions:

GeneratedParameters->None works with IncludeQuantities to allow mixtures of QuantityVariable and Quantity objects:

IncludeQuantities  (2)

Include additional constants and Quantity objects in the result:

Use the setting "PhysicalConstants" to include a standard set of physical constants:

Applications  (4)

Find the missing physical constants in the formula E^2 - p^2 == m^2:

Solve for the value of the constants:

Insert the correct exponents:

Eliminate unnecessary constants:

Find the dimensions of the constant needed to balance Kleiber's law :

Solve for the value of the mass exponent:

Estimate the power of a bomb blast by using only these physical quantities:

Construct a dimensionless combination:

Given the values of the parameters at a given time, estimate the energy of an explosion:

Determine possible dimensionless price impact functions depending on stock price, size and cost of bets, trading volumes and the volatility of the stock:

Find the general dimensionless combination:

Determine specific instances:

Properties & Relations  (1)

Formulas for dimensionless constants can be constructed from physical quantities:

Possible Issues  (5)

Only valid physical quantities can be used:

Dimensionless quantities will be omitted from the result:

Only valid constants will be used:

Angular units and physical quantities are not treated as dimensionless:

While returned combinations are dimensionless, they do not necessarily have a magnitude of one:

Interactive Examples  (1)

Examine all possible dimensionless combinations for a set of physical quantities and constants:

Neat Examples  (2)

Explore the possible dimensionless combinations of electromagnetic physical quantities:

Derive the factor for the fine structure constant from physical quantities:

Introduced in 2014
 (10.0)
 |
Updated in 2018
 (11.3)