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Nondimensionalization is the partial or full removal of physical dimensions from an
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
involving physical quantities by a suitable
substitution of variables Substitution may refer to: Arts and media *Chord substitution, in music, swapping one chord for a related one within a chord progression *Substitution (poetry), a variation in poetic scansion * "Substitution" (song), a 2009 song by Silversun Pic ...
. This technique can simplify and parameterize problems where measured units are involved. It is closely related to
dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (suc ...
. In some physical systems, the term scaling is used interchangeably with ''nondimensionalization'', in order to suggest that certain quantities are better measured relative to some appropriate unit. These units refer to quantities
intrinsic In science and engineering, an intrinsic property is a property of a specified subject that exists itself or within the subject. An extrinsic property is not essential or inherent to the subject that is being characterized. For example, m ...
to the system, rather than units such as SI units. Nondimensionalization is not the same as converting extensive quantities in an equation to intensive quantities, since the latter procedure results in variables that still carry units. Nondimensionalization can also recover characteristic properties of a system. For example, if a system has an intrinsic resonance frequency, length, or
time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a sy ...
, nondimensionalization can recover these values. The technique is especially useful for systems that can be described by
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
s. One important use is in the analysis of
control system A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial c ...
s. One of the simplest characteristic units is the
doubling time The doubling time is the time it takes for a population to double in size/value. It is applied to population growth, inflation, resource extraction, consumption of goods, compound interest, the volume of malignant tumours, and many other things th ...
of a system experiencing
exponential growth Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...
, or conversely the
half-life Half-life (symbol ) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable at ...
of a system experiencing
exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate c ...
; a more natural pair of characteristic units is mean age/
mean lifetime A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
, which correspond to base ''e'' rather than base 2. Many illustrative examples of nondimensionalization originate from simplifying differential equations. This is because a large body of physical problems can be formulated in terms of differential equations. Consider the following: * List of dynamical systems and differential equations topics *
List of partial differential equation topics A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby union ...
*
Differential equations of mathematical physics Differential may refer to: Mathematics * Differential (mathematics) comprises multiple related meanings of the word, both in calculus and differential geometry, such as an infinitesimal change in the value of a function * Differential algebra * ...
Although nondimensionalization is well adapted for these problems, it is not restricted to them. An example of a non-differential-equation application is dimensional analysis; another example is
normalization Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to: * Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of ...
in statistics.
Measuring device A measuring instrument is a device to measure a physical quantity. In the physical sciences, quality assurance, and engineering, measurement is the activity of obtaining and comparing physical quantities of real-world objects and events. Esta ...
s are practical examples of nondimensionalization occurring in everyday life. Measuring devices are calibrated relative to some known unit. Subsequent measurements are made relative to this standard. Then, the absolute value of the measurement is recovered by scaling with respect to the standard.

# Rationale

Suppose a
pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the ...
is swinging with a particular period ''T''. For such a system, it is advantageous to perform calculations relating to the swinging relative to ''T''. In some sense, this is normalizing the measurement with respect to the period. Measurements made relative to an intrinsic property of a system will apply to other systems which also have the same intrinsic property. It also allows one to compare a common property of different implementations of the same system. Nondimensionalization determines in a systematic manner the characteristic units of a system to use, without relying heavily on prior knowledge of the system's intrinsic properties (one should not confuse characteristic units of a ''system'' with natural units of ''nature''). In fact, nondimensionalization can suggest the parameters which should be used for analyzing a system. However, it is necessary to start with an equation that describes the system appropriately.

# Nondimensionalization steps

To nondimensionalize a system of equations, one must do the following: #Identify all the independent and dependent variables; #Replace each of them with a quantity scaled relative to a characteristic unit of measure to be determined; #Divide through by the coefficient of the highest order polynomial or derivative term; #Choose judiciously the definition of the characteristic unit for each variable so that the coefficients of as many terms as possible become 1; #Rewrite the system of equations in terms of their new dimensionless quantities. The last three steps are usually specific to the problem where nondimensionalization is applied. However, almost all systems require the first two steps to be performed.

## Conventions

There are no restrictions on the variable names used to replace "''x''" and "''t''". However, they are generally chosen so that it is convenient and intuitive to use for the problem at hand. For example, if "''x''" represented mass, the letter "''m''" might be an appropriate symbol to represent the dimensionless mass quantity. In this article, the following conventions have been used: * ''t'' – represents the independent variable – usually a time quantity. Its nondimensionalized counterpart is $\tau$. * ''x'' – represents the dependent variable – can be mass, voltage, or any measurable quantity. Its nondimensionalized counterpart is $\chi$. A subscripted ''c'' added to a quantity's variable name is used to denote the characteristic unit used to scale that quantity. For example, if ''x'' is a quantity, then ''xc'' is the characteristic unit used to scale it. As an illustrative example, consider a first order differential equation with
constant coefficients In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = ...
: $a\frac + bx = Af(t).$ # In this equation the independent variable here is ''t'', and the dependent variable is ''x''. # Set $x = \chi x_c, \ t = \tau t_c$. This results in the equation $a \frac \frac + b x_c \chi = A f(\tau t_c) \ \stackrel\ A F(\tau).$ # The coefficient of the highest ordered term is in front of the first derivative term. Dividing by this gives $\frac + \frac \chi = \frac F(\tau).$ # The coefficient in front of $\chi$ only contains one characteristic variable ''tc'', hence it is easiest to choose to set this to unity first: $\frac = 1 \Rightarrow t_c = \frac.$ Subsequently, $\frac = \frac = 1 \Rightarrow x_c = \frac.$ # The final dimensionless equation in this case becomes completely independent of any parameters with units: $\frac + \chi = F(\tau).$

## Substitutions

Suppose for simplicity that a certain system is characterized by two variables - a dependent variable ''x'' and an independent variable ''t'', where ''x'' is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orien ...
of ''t''. Both ''x'' and ''t'' represent quantities with units. To scale these two variables, assume there are two intrinsic units of measurement ''x''c and ''t''c with the same units as ''x'' and ''t'' respectively, such that these conditions hold: $\tau = \frac \Rightarrow t = \tau t_c$ $\chi = \frac \Rightarrow x = \chi x_c.$ These equations are used to replace ''x'' and ''t'' when nondimensionalizing. If differential operators are needed to describe the original system, their scaled counterparts become dimensionless differential operators.

### Differential operators

Consider the relationship $\,\! t = \tau t_c \Rightarrow dt = t_c d\tau \Rightarrow \frac = \frac.$ The dimensionless differential operators with respect to the independent variable becomes $\frac = \frac \frac = \frac \frac \Rightarrow \frac = \left( \frac \right)^n = \left( \frac \frac \right)^n = \frac \frac.$

### Forcing function

If a system has a forcing function $\,\! f\left(t\right)$ then $\,\! f(t) = f(\tau t_c) = f(t(\tau)) = F(\tau).$ Hence, the new forcing function $\,\! F$ is made to be dependent on the dimensionless quantity $\,\! \tau$.

# Linear differential equations with constant coefficients

## First order system

Consider the differential equation for a first order system: $a\frac + bx = Af(t).$ The derivation of the characteristic units for this system gives $t_c = \frac, \ x_c = \frac.$

## Second order system

A second order system has the form $a \frac + b\frac + cx = A f(t).$

### Substitution step

Replace the variables ''x'' and ''t'' with their scaled quantities. The equation becomes $a \frac \frac + b \frac \frac + c x_c \chi = A f(\tau t_c) = A F(\tau) .$ This new equation is not dimensionless, although all the variables with units are isolated in the coefficients. Dividing by the coefficient of the highest ordered term, the equation becomes $\frac + t_c \frac \frac + t_c^2 \frac \chi = \frac F(\tau).$ Now it is necessary to determine the quantities of ''x''''c'' and ''t''''c'' so that the coefficients become normalized. Since there are two free parameters, at most only two coefficients can be made to equal unity.

### Determination of characteristic units

Consider the variable ''t''''c'': #If $t_c = \frac$ the first order term is normalized. #If $t_c = \sqrt$ the zeroth order term is normalized. Both substitutions are valid. However, for pedagogical reasons, the latter substitution is used for second order systems. Choosing this substitution allows ''x''''c'' to be determined by normalizing the coefficient of the forcing function: $1 = \frac = \frac \Rightarrow x_c = \frac.$ The differential equation becomes $\frac + \frac \frac + \chi = F(\tau).$ The coefficient of the first order term is unitless. Define $2 \zeta \ \stackrel\ \frac.$ The factor 2 is present so that the solutions can be parameterized in terms of ''ζ''. In the context of mechanical or electrical systems, ''ζ'' is known as the
damping ratio Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples in ...
, and is an important parameter required in the analysis of
control system A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial c ...
s. 2''ζ'' is also known as the linewidth of the system. The result of the definition is the universal oscillator equation. $\frac + 2 \zeta \frac + \chi = F(\tau) .$

## Higher order systems

The general n-th order linear differential equation with constant coefficients has the form: $a_n \frac + a_ \frac + \ldots + a_1 \frac + a_0 x(t) = \sum_^n a_k \frac = Af(t).$ The function ''f''(''t'') is known as the forcing function. If the differential equation only contains real (not complex) coefficients, then the properties of such a system behaves as a mixture of first and second order systems only. This is because the roots of its
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The cha ...
are either real, or complex conjugate pairs. Therefore, understanding how nondimensionalization applies to first and second ordered systems allows the properties of higher order systems to be determined through superposition. The number of free parameters in a nondimensionalized form of a system increases with its order. For this reason, nondimensionalization is rarely used for higher order differential equations. The need for this procedure has also been reduced with the advent of
symbolic computation In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressio ...
.

## Examples of recovering characteristic units

A variety of systems can be approximated as either first or second order systems. These include mechanical, electrical, fluidic, caloric, and torsional systems. This is because the fundamental physical quantities involved within each of these examples are related through first and second order derivatives.

### Mechanical oscillations

Suppose we have a mass attached to a spring and a damper, which in turn are attached to a wall, and a force acting on the mass along the same line. Define * $x$ = displacement from equilibrium * $t$ = time * $f$ = external force or "disturbance" applied to system g⋅m⋅s−2* $m$ = mass of the block g* $B$ = damping constant of dashpot g⋅s−1* $k$ = force constant of spring g⋅s−2 Suppose the applied force is a sinusoid , the differential equation that describes the motion of the block is $m \frac + B \frac + kx = F_0 \cos(\omega t)$ Nondimensionalizing this equation the same way as described under second order system yields several characteristics of the system. The intrinsic unit ''xc'' corresponds to the distance the block moves per unit force $x_c = \frac.$ The characteristic variable ''tc'' is equal to the period of the oscillations $t_c = \sqrt$ and the dimensionless variable 2''ζ'' corresponds to the linewidth of the system. ''ζ'' itself is the
damping ratio Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples in ...
. $2 \zeta = \frac$

### = First-order series RC circuit

= For a series RC attached to a
voltage source A voltage source is a two- terminal device which can maintain a fixed voltage. An ideal voltage source can maintain the fixed voltage independent of the load resistance or the output current. However, a real-world voltage source cannot supply u ...
$R \frac + \frac = V(t) \Rightarrow \frac + \chi = F(\tau)$ with substitutions $Q = \chi x_c, \ t = \tau t_c, \ x_c = C V_0, \ t_c = RC, \ F = V.$ The first characteristic unit corresponds to the total
charge Charge or charged may refer to: Arts, entertainment, and media Films * '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * '' Charge!!'', an album by The Aq ...
in the circuit. The second characteristic unit corresponds to the
time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a sy ...
for the system.

### = Second-order series RLC circuit

= For a series configuration of ''R'',''C'',''L'' components where ''Q'' is the charge in the system $L \frac + R \frac + \frac = V_0 \cos(\omega t) \Rightarrow \frac + 2 \zeta \frac + \chi = \cos(\Omega \tau)$ with the substitutions $Q = \chi x_c, \ t = \tau t_c, \ \ x_c = C V_0, \ t_c = \sqrt, \ 2 \zeta = R \sqrt, \ \Omega = t_c \omega.$ The first variable corresponds to the maximum charge stored in the circuit. The resonance frequency is given by the reciprocal of the characteristic time. The last expression is the linewidth of the system. The Ω can be considered as a normalized forcing function frequency.

## Quantum mechanics

### Quantum harmonic oscillator

The
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
for the one-dimensional time independent
quantum harmonic oscillator 量子調和振動子 は、 古典調和振動子 の 量子力学 類似物です。任意の滑らかな ポテンシャル は通常、安定した 平衡点 の近くで 調和ポテンシャル として近似できるため、最� ...
is $\left(-\frac \frac + \fracm \omega^2 x^2\right) \psi(x) = E \psi(x).$ The modulus square of the wavefunction represents probability density that, when integrated over , gives a dimensionless probability. Therefore, has units of inverse length. To nondimensionalize this, it must be rewritten as a function of a dimensionless variable. To do this, we substitute $\tilde x \equiv \frac,$ where is some characteristic length of this system. This gives us a dimensionless wave function $\tilde \psi$ defined via $\psi(x) = \psi(\tilde x x_) = \psi(x(x_)) = \tilde \psi(\tilde x).$ The differential equation then becomes $\left(-\frac \frac \frac + \frac m \omega^2 x_^2 \tilde x^2 \right) \tilde \psi(\tilde x) = E \, \tilde \psi(\tilde x) \Rightarrow \left(-\frac + \frac \tilde x^2 \right) \tilde \psi(\tilde x) = \frac \tilde \psi(\tilde x).$ To make the term in front of $\tilde x^2$ dimensionless, set $\frac = 1 \Rightarrow x_ = \sqrt .$ The fully nondimensionalized equation is $\left(-\frac + \tilde x^2 \right) \tilde \psi(\tilde x) = \tilde E \tilde \psi(\tilde x),$ where we have defined $E \equiv \frac \tilde E.$ The factor in front of $\tilde E$ is in fact (coincidentally) the
ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
energy of the harmonic oscillator. Usually, the energy term is not made dimensionless as we are interested in determining the energies of the quantum states. Rearranging the first equation, the familiar equation for the harmonic oscillator becomes $\frac \left( -\frac + \tilde x^2 \right) \tilde \psi(\tilde x) = E \tilde \psi(\tilde x).$

# Statistical analogs

In statistics, the analogous process is usually dividing a difference (a distance) by a scale factor (a measure of
statistical dispersion In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquart ...
), which yields a dimensionless number, which is called ''
normalization Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to: * Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of ...
.'' Most often, this is dividing errors or residuals by the
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, w ...
or sample standard deviation, respectively, yielding standard scores and
studentized residual In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. It is a form of a Student's ''t''-statistic, with the estimate of error varying between points. This i ...
s.

*
Buckingham π theorem In engineering, applied mathematics, and physics, the Buckingham theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physical ...
* Dimensionless number * Natural units * System equivalence *
RLC circuit An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components ...
*
RL circuit A resistor–inductor circuit (RL circuit), or RL filter or RL network, is an electric circuit composed of resistors and inductors driven by a voltage or current source. A first-order RL circuit is composed of one resistor and one inductor, either ...
*
RC circuit A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC ...
* Logistic equation {{colend