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This technique can simplify and [[parametric equation|parameterize]] problems where [[Measurement|measured]] units are involved. It is closely related to [[dimensional analysis]]. In some physical [[system]]s, 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 [[Wiktionary:intrinsic|intrinsic]] to the system, rather than units such as [[International System of Units|SI]] units. Nondimensionalization is not the same as converting [[intensive and extensive properties|extensive quantities]] in an equation to intensive quantities, since the latter procedure results in variables that still carry units.<ref>{{Cite web |title=How does non-dimensionalization improve the behavior of ODE solvers? |url=https://scicomp.stackexchange.com/questions/41600/how-does-non-dimensionalization-improve-the-behavior-of-ode-solvers |access-date=2024-08-23 |website=Computational Science Stack Exchange |language=en}}</ref> Nondimensionalization can also recover characteristic properties of a system. For example, if a system has an intrinsic [[Resonance|resonance frequency]], [[length]], or [[time constant]], nondimensionalization can recover these values. The technique is especially useful for systems that can be described by [[differential equation]]s. One important use is in the analysis of [[control system]]s. One of the simplest characteristic units is the [[doubling time]] of a system experiencing [[exponential growth]], or conversely the [[half-life]] of a system experiencing [[exponential decay]]; a more natural pair of characteristic units is mean age/[[Exponential_decay#Mean_lifetime|mean lifetime]], 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]] * [[Differential equations of mathematical physics]] 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 (statistics)|normalization]] in [[statistics]]. [[Measuring instrument|Measuring device]]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]] is swinging with a particular [[Frequency|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 <math>\tau</math>. * ''x'' – represents the dependent variable – can be mass, voltage, or any measurable quantity. Its nondimensionalized counterpart is <math>\chi</math>. A subscript '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 ''x''<sub>c</sub> is the characteristic unit used to scale it. As an illustrative example, consider a first order differential equation with [[constant coefficients]]: <math display="block">a\frac{dx}{dt} + bx = Af(t).</math> # In this equation the independent variable here is ''t'', and the dependent variable is ''x''. # Set <math>x = \chi x_\text{c}, \ t = \tau t_\text{c}</math>. This results in the equation <math display="block">a \frac{x_\text{c}}{t_\text{c}} \frac{d \chi}{d \tau} + b x_\text{c} \chi = A f(\tau t_\text{c}) \ \stackrel{\mathrm{def}}{=}\ A F(\tau).</math> # The coefficient of the highest ordered term is in front of the first derivative term. Dividing by this gives <math display="block">\frac{d \chi}{d \tau} + \frac{b t_\text{c}}{a} \chi = \frac{A t_\text{c}}{a x_\text{c}} F(\tau).</math> # The coefficient in front of <math>\chi</math> only contains one characteristic variable ''t''<sub>c</sub>, hence it is easiest to choose to set this to unity first: {{NumBlk|:::|<math>\frac{b t_\text{c}}{a} = 1 \Rightarrow t_\text{c} = \frac{a}{b}.</math>|{{EqRef|1}}}} :: Subsequently, {{NumBlk|:::|<math>\frac{A t_\text{c}}{a x_\text{c}} = \frac{A}{b x_\text{c}} = 1 \Rightarrow x_\text{c} = \frac{A}{b}.</math>|{{EqRef|2}}}} # The final dimensionless equation in this case becomes completely independent of any parameters with units: <math display="block">\frac{d \chi}{d \tau} + \chi = F(\tau).</math> === 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 (mathematics)|function]] of ''t''. Both ''x'' and ''t'' represent quantities with units. To scale these two variables, assume there are two intrinsic units of measurement ''x''<sub>c</sub> and ''t''<sub>c</sub> with the same units as ''x'' and ''t'' respectively, such that these conditions hold: <math display="block">\tau = \frac{t}{t_\text{c}} \Rightarrow t = \tau t_\text{c} </math> <math display="block"> \chi = \frac{x}{x_\text{c}} \Rightarrow x = \chi x_\text{c}.</math> 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 <math display="block">t = \tau t_\text{c} \Rightarrow dt = t_\text{c} d\tau \Rightarrow \frac{d\tau}{dt} = \frac{1}{t_\text{c}}.</math> The dimensionless differential operators with respect to the independent variable becomes <math display="block">\frac{d}{dt} = \frac{d \tau}{dt} \frac{d}{d \tau} = \frac{1}{t_\text{c}} \frac{d}{d \tau} \Rightarrow \frac{d^n}{dt^n} = \left( \frac{d}{dt} \right)^n = \left( \frac{1}{t_\text{c}} \frac{d}{d \tau} \right)^n = \frac{1}{{t_\text{c}}^n} \frac{d^n}{d \tau^n}.</math> ==== Forcing function ==== If a system has a [[forcing function (differential equations)|forcing function]] <math>f(t)</math> then <math display="block">f(t) = f(\tau t_\text{c}) = f(t(\tau)) = F(\tau).</math> Hence, the new forcing function <math> F </math> is made to be dependent on the dimensionless quantity <math>\tau </math>. == Linear differential equations with constant coefficients == === First order system === Consider the differential equation for a first order system: <math display="block">a\frac{dx}{dt} + bx = Af(t).</math> The derivation of the characteristic units to {{EquationNote|1|Eq. 1}} and {{EquationNote|2|Eq. 2}} for this system gave <math display="block">t_\text{c} = \frac{a}{b}, \ x_\text{c} = \frac{A}{b}.</math> === Second order system === A second order system has the form <math display="block">a \frac{d^2 x}{dt^2} + b\frac{dx}{dt} + cx = A f(t).</math> ==== Substitution step ==== Replace the variables ''x'' and ''t'' with their scaled quantities. The equation becomes <math display="block">a \frac{x_\text{c}}{{t_\text{c}}^2} \frac{ d^2 \chi}{d \tau^2} + b \frac{x_\text{c}}{t_\text{c}} \frac{d \chi}{d \tau} + c x_\text{c} \chi = A f(\tau t_\text{c}) = A F(\tau) .</math> 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 <math display="block"> \frac{d^2 \chi}{d \tau^2} + t_\text{c} \frac{b}{a} \frac{d \chi}{d \tau} + {t_\text{c}}^2 \frac{c}{a} \chi = \frac{A {t_\text{c}}^2}{a x_\text{c}} F(\tau).</math> Now it is necessary to determine the quantities of ''x''<sub>c</sub> and ''t''<sub>c</sub> 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''<sub>c</sub>: #If <math> t_\text{c} = \frac{a}{b} </math> the first order term is normalized. #If <math> t_\text{c} = \sqrt{\frac{a}{c}} </math> 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''<sub>c</sub> to be determined by normalizing the coefficient of the forcing function: <math display="block">1 = \frac{A t_\text{c}^2}{a x_\text{c}} = \frac{A}{c x_\text{c}} \Rightarrow x_\text{c} = \frac{A}{c}.</math> The differential equation becomes <math display="block">\frac{d^2 \chi}{d \tau^2} + \frac{b}{\sqrt{ac}} \frac{d \chi}{d\tau} + \chi = F(\tau). </math> The coefficient of the first order term is unitless. Define <math display="block">2 \zeta \ \stackrel{\mathrm{def}}{=}\ \frac{b}{\sqrt{ac}}. </math> 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]], and is an important parameter required in the analysis of [[control system]]s. 2''ζ'' is also known as the [[linewidth]] of the system. The result of the definition is the [[Harmonic oscillator#Universal oscillator equation|universal oscillator equation]]. <math display="block">\frac{d^2 \chi}{d \tau^2} + 2 \zeta \frac{d \chi}{d\tau} + \chi = F(\tau) .</math> === Higher order systems === The general ''n''th order linear differential equation with constant coefficients has the form: <math display="block">a_n \frac{d^n}{dt^n} x(t) + a_{n-1} \frac{d^{n-1}}{dt^{n-1}} x(t) + \ldots + a_1 \frac{d}{dt} x(t) + a_0 x(t) = \sum_{k = 0}^n a_k \big( \frac{d}{dt} \big) ^k x(t) = Af(t). </math> The function ''f''(''t'') is known as the [[forcing function (differential equations)|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 [[root of a function|roots]] of its [[characteristic polynomial]] are either [[Real number|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 principle|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]]. === 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 ==== [[Image:Mass-Spring-Damper.png|thumb|300px|A mass attached to a spring and a damper.]] 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 * <math>x</math> = displacement from equilibrium [m] * <math>t</math> = time [s] * <math>f</math> = external force or "disturbance" applied to system [kg⋅m⋅s<sup>−2</sup>] * <math>m</math> = mass of the block [kg] * <math>B</math> = damping constant of dashpot [kg⋅s<sup>−1</sup>] * <math>k</math> = force constant of spring [kg⋅s<sup>−2</sup>] Suppose the applied force is a sinusoid {{nowrap|1=''F'' = ''F''<sub>0</sub> cos(''ωt'')}}, the differential equation that describes the motion of the block is <math display="block">m \frac{d^2 x}{d t^2} + B \frac{d x}{d t} + kx = F_0 \cos(\omega t)</math> Nondimensionalizing this equation the same way as described under {{slink||Second order system}} yields several characteristics of the system: * The intrinsic unit ''x''<sub>c</sub> corresponds to the distance the block moves per unit force <math display="block">x_\text{c} = \frac{F_0}{k}.</math> * The characteristic variable ''t''<sub>c</sub> is equal to the period of the oscillations <math display="block">t_\text{c} = \sqrt{\frac{m}{k}}</math> * The dimensionless variable 2''ζ'' corresponds to the linewidth of the system. <math display="block">2 \zeta = \frac{B}{\sqrt{mk}}</math> * ''ζ'' itself is the [[damping ratio]] ==== Electrical oscillations ==== ===== First-order series RC circuit ===== For a series [[RC circuit|RC]] attached to a [[power supply|voltage source]] <math display="block">R \frac{dQ}{dt} + \frac{Q}{C} = V(t) \Rightarrow \frac{d \chi}{d \tau} + \chi = F(\tau)</math> with substitutions <math display="block">Q = \chi x_\text{c}, \ t = \tau t_\text{c}, \ x_\text{c} = C V_0, \ t_\text{c} = RC, \ F = V.</math> The first characteristic unit corresponds to the total [[electric charge|charge]] in the circuit. The second characteristic unit corresponds to the [[time constant]] 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 <math display="block"> L \frac{d^2 Q}{dt^2} + R \frac{d Q}{d t} + \frac{Q}{C} = V_0 \cos(\omega t) \Rightarrow \frac{d^2 \chi}{d \tau^2} + 2 \zeta \frac{d \chi}{d\tau} + \chi = \cos(\Omega \tau) </math> with the substitutions <math display="block">Q = \chi x_\text{c}, \ t = \tau t_\text{c}, \ \ x_\text{c} = C V_0, \ t_\text{c} = \sqrt{LC}, \ 2 \zeta = R \sqrt{\frac{C}{L}}, \ \Omega = t_\text{c} \omega.</math> 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]] for the one-dimensional time independent [[quantum harmonic oscillator]] is <math display="block">\left(-\frac{\hbar^2}{2m} \frac{d^2}{d x^2} + \frac{1}{2}m \omega^2 x^2\right) \psi(x) = E \psi(x).</math> The modulus square of the [[wavefunction]] {{math|{{!}}''ψ''(''x''){{!}}<sup>2</sup>}} represents probability density that, when integrated over {{math|''x''}}, gives a dimensionless probability. Therefore, {{math|{{!}}''ψ''(''x''){{!}}<sup>2</sup>}} has units of inverse length. To nondimensionalize this, it must be rewritten as a function of a dimensionless variable. To do this, we substitute <math display="block">\tilde x \equiv \frac{x}{x_{\text{c}}},</math> where {{math|''x''<sub>c</sub>}} is some characteristic length of this system. This gives us a dimensionless wave function <math>\tilde \psi</math> defined via <math display="block">\psi(x) = \psi(\tilde x x_{\text{c}}) = \psi(x(x_{\text{c}})) = \tilde \psi(\tilde x).</math> The differential equation then becomes <math display="block">\left(-\frac{\hbar^2}{2m} \frac{1}{x_{\text{c}}^2} \frac{d^2}{d \tilde x^2} + \frac{1}{2} m \omega^2 x_{\text{c}}^2 \tilde x^2 \right) \tilde \psi(\tilde x) = E \, \tilde \psi(\tilde x) \Rightarrow \left(-\frac{d^2}{d \tilde x^2} + \frac{m^2 \omega^2 x_{\text{c}}^4}{\hbar^2} \tilde x^2 \right) \tilde \psi(\tilde x) = \frac{2 m x_{\text{c}}^2 E}{\hbar^2} \tilde \psi(\tilde x).</math> To make the term in front of <math>\tilde x^2</math> dimensionless, set <math display="block">\frac{m^2 \omega^2 x_{\text{c}}^4}{\hbar^2} = 1 \Rightarrow x_{\text{c}} = \sqrt{\frac{\hbar}{m \omega}} . </math> The fully nondimensionalized equation is <math display="block">\left(-\frac{d^2}{d \tilde x^2} + \tilde x^2 \right) \tilde \psi(\tilde x) = \tilde E \tilde \psi(\tilde x),</math> where we have defined <math display="block">E \equiv \frac{\hbar \omega}{2} \tilde E.</math> The factor in front of <math>\tilde E</math> is in fact (coincidentally) 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 state]]s. Rearranging the first equation, the familiar equation for the harmonic oscillator becomes <math display="block">\frac{\hbar \omega}{2} \left( -\frac{d^2}{d \tilde x^2} + \tilde x^2 \right) \tilde \psi(\tilde x) = E \tilde \psi(\tilde x).</math>        == Statistical analogs == {{main|Normalization (statistics)}} In [[statistics]], the analogous process is usually dividing a difference (a distance) by a scale factor (a measure of [[statistical dispersion]]), which yields a dimensionless number, which is called ''[[Normalization (statistics)|normalization]].'' Most often, this is dividing [[errors and residuals in statistics|errors or residuals]] by the [[standard deviation]] or sample standard deviation, respectively, yielding [[standard score]]s and [[studentized residual]]s. == See also == {{colbegin}} * [[Buckingham π theorem]] * [[Dimensionless number]] * [[Natural units]] * [[System equivalence]] * [[RLC circuit]] * [[RL circuit]] * [[RC circuit]] * [[Logistic map|Logistic equation]] * [[Per-unit system]] {{colend}} == References == {{Reflist}} == External links== * [http://www.royalsociety.org.nz/publications/journals/nzja/1998/059/ Analysis of differential equation models in biology: a case study for clover meristem populations] (Application of nondimensionalization to a problem in biology). * [https://web.archive.org/web/20050306141857/http://www.maths.bath.ac.uk/~masjde/MSc/CourseNotes/MA50176.pdf Course notes for Mathematical Modelling and Industrial Mathematics] ''Jonathan Evans, Department of Mathematical Sciences, [[University of Bath]]''. (see Chapter 3). * [https://hplgit.github.io/scaling-book/doc/pub/book/pdf/scaling-book-4screen-sol.pdf Scaling of Differential Equations] ''Hans Petter Langtangen, Geir K. Pedersen, Center for Biomedical Computing, Simula Research Laboratory and Department of Informatics, [[University of Oslo]]''. 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