# 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, "PhysicalQuantity" entities 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:
•  GeneratedParameters C how 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(3)

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:

"PhysicalQuantity" entities, including in QuantityVariable expressions, can also be used:

## Options(5)

### GeneratedParameters(3)

Use a different symbol for parameters:

By default, a generic solution is returned:

Use to get specific solutions:

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: