Nondimensionalization

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class="vector-toc-numb">2</span> <span>Nondimensionalization steps</span> </div> </a> <button aria-controls="toc-Nondimensionalization_steps-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Nondimensionalization steps subsection</span> </button> <ul id="toc-Nondimensionalization_steps-sublist" class="vector-toc-list"> <li id="toc-Conventions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conventions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Conventions</span> </div> </a> <ul id="toc-Conventions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Substitutions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Substitutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Substitutions</span> </div> </a> <ul id="toc-Substitutions-sublist" class="vector-toc-list"> <li id="toc-Differential_operators" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Differential_operators"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Differential operators</span> </div> </a> <ul id="toc-Differential_operators-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Forcing_function" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Forcing_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Forcing function</span> </div> </a> <ul id="toc-Forcing_function-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Linear_differential_equations_with_constant_coefficients" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Linear_differential_equations_with_constant_coefficients"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Linear differential equations with constant coefficients</span> </div> </a> <button aria-controls="toc-Linear_differential_equations_with_constant_coefficients-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Linear differential equations with constant coefficients subsection</span> </button> <ul id="toc-Linear_differential_equations_with_constant_coefficients-sublist" class="vector-toc-list"> <li id="toc-First_order_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#First_order_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>First order system</span> </div> </a> <ul id="toc-First_order_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Second_order_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Second_order_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Second order system</span> </div> </a> <ul id="toc-Second_order_system-sublist" class="vector-toc-list"> <li id="toc-Substitution_step" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Substitution_step"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Substitution step</span> </div> </a> <ul id="toc-Substitution_step-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Determination_of_characteristic_units" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Determination_of_characteristic_units"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Determination of characteristic units</span> </div> </a> <ul id="toc-Determination_of_characteristic_units-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Higher_order_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Higher_order_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Higher order systems</span> </div> </a> <ul id="toc-Higher_order_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples_of_recovering_characteristic_units" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples_of_recovering_characteristic_units"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Examples of recovering characteristic units</span> </div> </a> <ul id="toc-Examples_of_recovering_characteristic_units-sublist" class="vector-toc-list"> <li id="toc-Mechanical_oscillations" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Mechanical_oscillations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4.1</span> <span>Mechanical oscillations</span> </div> </a> <ul id="toc-Mechanical_oscillations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electrical_oscillations" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Electrical_oscillations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4.2</span> <span>Electrical oscillations</span> </div> </a> <ul id="toc-Electrical_oscillations-sublist" class="vector-toc-list"> <li id="toc-First-order_series_RC_circuit" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#First-order_series_RC_circuit"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4.2.1</span> <span>First-order series RC circuit</span> </div> </a> <ul id="toc-First-order_series_RC_circuit-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Second-order_series_RLC_circuit" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Second-order_series_RLC_circuit"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4.2.2</span> <span>Second-order series RLC circuit</span> </div> </a> <ul id="toc-Second-order_series_RLC_circuit-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Quantum_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Quantum mechanics</span> </div> </a> <ul id="toc-Quantum_mechanics-sublist" class="vector-toc-list"> <li id="toc-Quantum_harmonic_oscillator" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Quantum_harmonic_oscillator"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5.1</span> <span>Quantum harmonic oscillator</span> </div> </a> <ul id="toc-Quantum_harmonic_oscillator-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Statistical_analogs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Statistical_analogs"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Statistical analogs</span> </div> </a> <ul id="toc-Statistical_analogs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" 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data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Characteristic_units&redirect=no" class="mw-redirect" title="Characteristic units">Characteristic units</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical simplification technique in physical sciences</div><p><b>Nondimensionalization</b> is the partial or full removal of <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">physical dimensions</a> from an <a href="/wiki/Mathematical_equation" class="mw-redirect" title="Mathematical equation">equation</a> involving <a href="/wiki/Physical_quantity" title="Physical quantity">physical quantities</a> by a suitable <a href="/wiki/Substitution_of_variables" class="mw-redirect" title="Substitution of variables">substitution of variables</a>. This technique can simplify and <a href="/wiki/Parametric_equation" title="Parametric equation">parameterize</a> problems where <a href="/wiki/Measurement" title="Measurement">measured</a> units are involved. It is closely related to <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimensional analysis</a>. In some physical <a href="/wiki/System" title="System">systems</a>, the term <i><b>scaling</b></i> is used interchangeably with <i>nondimensionalization</i>, in order to suggest that certain quantities are better measured relative to some appropriate unit. These units refer to quantities <a href="https://en.wiktionary.org/wiki/intrinsic" class="extiw" title="wiktionary:intrinsic">intrinsic</a> to the system, rather than units such as <a href="/wiki/International_System_of_Units" title="International System of Units">SI</a> units. Nondimensionalization is not the same as converting <a href="/wiki/Intensive_and_extensive_properties" title="Intensive and extensive properties">extensive quantities</a> in an equation to intensive quantities, since the latter procedure results in variables that still carry units.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Nondimensionalization can also recover characteristic properties of a system. For example, if a system has an intrinsic <a href="/wiki/Resonance" title="Resonance">resonance frequency</a>, <a href="/wiki/Length" title="Length">length</a>, or <a href="/wiki/Time_constant" title="Time constant">time constant</a>, nondimensionalization can recover these values. The technique is especially useful for systems that can be described by <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>. One important use is in the analysis of <a href="/wiki/Control_system" title="Control system">control systems</a>. One of the simplest characteristic units is the <a href="/wiki/Doubling_time" title="Doubling time">doubling time</a> of a system experiencing <a href="/wiki/Exponential_growth" title="Exponential growth">exponential growth</a>, or conversely the <a href="/wiki/Half-life" title="Half-life">half-life</a> of a system experiencing <a href="/wiki/Exponential_decay" title="Exponential decay">exponential decay</a>; a more natural pair of characteristic units is mean age/<a href="/wiki/Exponential_decay#Mean_lifetime" title="Exponential decay">mean lifetime</a>, which correspond to base <i>e</i> rather than base 2. </p><p>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: </p> <ul><li><a href="/wiki/List_of_dynamical_systems_and_differential_equations_topics" title="List of dynamical systems and differential equations topics">List of dynamical systems and differential equations topics</a></li> <li><a href="/wiki/List_of_partial_differential_equation_topics" title="List of partial differential equation topics">List of partial differential equation topics</a></li> <li><a href="/wiki/Differential_equations_of_mathematical_physics" class="mw-redirect" title="Differential equations of mathematical physics">Differential equations of mathematical physics</a></li></ul> <p>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 <a href="/wiki/Normalization_(statistics)" title="Normalization (statistics)">normalization</a> in <a href="/wiki/Statistics" title="Statistics">statistics</a>. </p><p><a href="/wiki/Measuring_instrument" class="mw-redirect" title="Measuring instrument">Measuring devices</a> 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. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Rationale">Rationale</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=1" title="Edit section: Rationale"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose a <a href="/wiki/Pendulum" title="Pendulum">pendulum</a> is swinging with a particular <a href="/wiki/Frequency" title="Frequency">period</a> <i>T</i>. For such a system, it is advantageous to perform calculations relating to the swinging relative to <i>T</i>. In some sense, this is normalizing the measurement with respect to the period. </p><p>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 <b>characteristic units</b> 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 <i>system</i> with <a href="/wiki/Natural_units" title="Natural units">natural units</a> of <i>nature</i>). 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. </p> <div class="mw-heading mw-heading2"><h2 id="Nondimensionalization_steps">Nondimensionalization steps</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=2" title="Edit section: Nondimensionalization steps"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To nondimensionalize a system of equations, one must do the following: </p> <ol><li>Identify all the independent and dependent variables;</li> <li>Replace each of them with a quantity scaled relative to a characteristic unit of measure to be determined;</li> <li>Divide through by the coefficient of the highest order polynomial or derivative term;</li> <li>Choose judiciously the definition of the characteristic unit for each variable so that the coefficients of as many terms as possible become 1;</li> <li>Rewrite the system of equations in terms of their new dimensionless quantities.</li></ol> <p>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. </p> <div class="mw-heading mw-heading3"><h3 id="Conventions">Conventions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=3" title="Edit section: Conventions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are no restrictions on the variable names used to replace "<i>x</i>" and "<i>t</i>". However, they are generally chosen so that it is convenient and intuitive to use for the problem at hand. For example, if "<i>x</i>" represented mass, the letter "<i>m</i>" might be an appropriate symbol to represent the dimensionless mass quantity. </p><p>In this article, the following conventions have been used: </p> <ul><li><i>t</i> – represents the independent variable – usually a time quantity. Its nondimensionalized counterpart is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span>.</li> <li><i>x</i> – represents the dependent variable – can be mass, voltage, or any measurable quantity. Its nondimensionalized counterpart is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }"></span>.</li></ul> <p>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 <i>x</i> is a quantity, then <i>x</i><sub>c</sub> is the characteristic unit used to scale it. </p><p>As an illustrative example, consider a first order differential equation with <a href="/wiki/Constant_coefficients" class="mw-redirect" title="Constant coefficients">constant coefficients</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\frac {dx}{dt}}+bx=Af(t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>=</mo> <mi>A</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\frac {dx}{dt}}+bx=Af(t).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b02f58af60f31b8d8e0c778b59aacc5619d6989" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.195ex; height:5.509ex;" alt="{\displaystyle a{\frac {dx}{dt}}+bx=Af(t).}"></span> </p> <ol><li>In this equation the independent variable here is <i>t</i>, and the dependent variable is <i>x</i>.</li> <li>Set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\chi x_{\text{c}},\ t=\tau t_{\text{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>χ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo>=</mo> <mi>τ</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\chi x_{\text{c}},\ t=\tau t_{\text{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c459c20a4ec06fd6625880e950613e76f75aea92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.732ex; height:2.343ex;" alt="{\displaystyle x=\chi x_{\text{c}},\ t=\tau t_{\text{c}}}"></span>. This results in the equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\frac {x_{\text{c}}}{t_{\text{c}}}}{\frac {d\chi }{d\tau }}+bx_{\text{c}}\chi =Af(\tau t_{\text{c}})\ {\stackrel {\mathrm {def} }{=}}\ AF(\tau ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>χ</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>b</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mi>χ</mi> <mo>=</mo> <mi>A</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mi>A</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\frac {x_{\text{c}}}{t_{\text{c}}}}{\frac {d\chi }{d\tau }}+bx_{\text{c}}\chi =Af(\tau t_{\text{c}})\ {\stackrel {\mathrm {def} }{=}}\ AF(\tau ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6aa6a097b63792b99e5e872df86010f0bb3bee2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:36.942ex; height:5.676ex;" alt="{\displaystyle a{\frac {x_{\text{c}}}{t_{\text{c}}}}{\frac {d\chi }{d\tau }}+bx_{\text{c}}\chi =Af(\tau t_{\text{c}})\ {\stackrel {\mathrm {def} }{=}}\ AF(\tau ).}"></span></li> <li>The coefficient of the highest ordered term is in front of the first derivative term. Dividing by this gives <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\chi }{d\tau }}+{\frac {bt_{\text{c}}}{a}}\chi ={\frac {At_{\text{c}}}{ax_{\text{c}}}}F(\tau ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>χ</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> <mi>a</mi> </mfrac> </mrow> <mi>χ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> <mrow> <mi>a</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\chi }{d\tau }}+{\frac {bt_{\text{c}}}{a}}\chi ={\frac {At_{\text{c}}}{ax_{\text{c}}}}F(\tau ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bd4aa6c7ca8005147a407197faeaa3042686245" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.317ex; height:5.676ex;" alt="{\displaystyle {\frac {d\chi }{d\tau }}+{\frac {bt_{\text{c}}}{a}}\chi ={\frac {At_{\text{c}}}{ax_{\text{c}}}}F(\tau ).}"></span></li> <li>The coefficient in front of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }"></span> only contains one characteristic variable <i>t</i><sub>c</sub>, hence it is easiest to choose to set this to unity first:</li></ol> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 4.8em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {bt_{\text{c}}}{a}}=1\Rightarrow t_{\text{c}}={\frac {a}{b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo stretchy="false">⇒</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {bt_{\text{c}}}{a}}=1\Rightarrow t_{\text{c}}={\frac {a}{b}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69202d861df712603a00177aeba422b77f4a9e0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.124ex; height:5.509ex;" alt="{\displaystyle {\frac {bt_{\text{c}}}{a}}=1\Rightarrow t_{\text{c}}={\frac {a}{b}}.}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span>)</b></td></tr></tbody></table> <dl><dd><dl><dd>Subsequently,</dd></dl></dd></dl> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 4.8em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {At_{\text{c}}}{ax_{\text{c}}}}={\frac {A}{bx_{\text{c}}}}=1\Rightarrow x_{\text{c}}={\frac {A}{b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> <mrow> <mi>a</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mrow> <mi>b</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo stretchy="false">⇒</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mi>b</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {At_{\text{c}}}{ax_{\text{c}}}}={\frac {A}{bx_{\text{c}}}}=1\Rightarrow x_{\text{c}}={\frac {A}{b}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/324ffada347b138059bb32ff772ca25de0fea001" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.097ex; height:5.843ex;" alt="{\displaystyle {\frac {At_{\text{c}}}{ax_{\text{c}}}}={\frac {A}{bx_{\text{c}}}}=1\Rightarrow x_{\text{c}}={\frac {A}{b}}.}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table> <ol><li>The final dimensionless equation in this case becomes completely independent of any parameters with units: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\chi }{d\tau }}+\chi =F(\tau ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>χ</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>χ</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\chi }{d\tau }}+\chi =F(\tau ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c2429bafd09ed8a8c4dd28d36b5549a22a08249" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.3ex; height:5.509ex;" alt="{\displaystyle {\frac {d\chi }{d\tau }}+\chi =F(\tau ).}"></span></li></ol> <div class="mw-heading mw-heading3"><h3 id="Substitutions">Substitutions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=4" title="Edit section: Substitutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose for simplicity that a certain system is characterized by two variables – a dependent variable <i>x</i> and an independent variable <i>t</i>, where <i>x</i> is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of <i>t</i>. Both <i>x</i> and <i>t</i> represent quantities with units. To scale these two variables, assume there are two intrinsic units of measurement <i>x</i><sub>c</sub> and <i>t</i><sub>c</sub> with the same units as <i>x</i> and <i>t</i> respectively, such that these conditions hold: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau ={\frac {t}{t_{\text{c}}}}\Rightarrow t=\tau t_{\text{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mfrac> </mrow> <mo stretchy="false">⇒</mo> <mi>t</mi> <mo>=</mo> <mi>τ</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau ={\frac {t}{t_{\text{c}}}}\Rightarrow t=\tau t_{\text{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d11908f16393f64139977a17a55a7c110fa7e19f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.494ex; height:5.509ex;" alt="{\displaystyle \tau ={\frac {t}{t_{\text{c}}}}\Rightarrow t=\tau t_{\text{c}}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ={\frac {x}{x_{\text{c}}}}\Rightarrow x=\chi x_{\text{c}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mfrac> </mrow> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mi>χ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi ={\frac {x}{x_{\text{c}}}}\Rightarrow x=\chi x_{\text{c}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/781deec8c6e618523a50002c9a63daa297ea2e71" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.118ex; height:5.009ex;" alt="{\displaystyle \chi ={\frac {x}{x_{\text{c}}}}\Rightarrow x=\chi x_{\text{c}}.}"></span> </p><p>These equations are used to replace <i>x</i> and <i>t</i> when nondimensionalizing. If differential operators are needed to describe the original system, their scaled counterparts become dimensionless differential operators. </p> <div class="mw-heading mw-heading4"><h4 id="Differential_operators">Differential operators</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=5" title="Edit section: Differential operators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the relationship <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=\tau t_{\text{c}}\Rightarrow dt=t_{\text{c}}d\tau \Rightarrow {\frac {d\tau }{dt}}={\frac {1}{t_{\text{c}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mi>τ</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo stretchy="false">⇒</mo> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mi>d</mi> <mi>τ</mi> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=\tau t_{\text{c}}\Rightarrow dt=t_{\text{c}}d\tau \Rightarrow {\frac {d\tau }{dt}}={\frac {1}{t_{\text{c}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2298d9b830f88808bc66b277bf3f9b28a3cd47d6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:33.181ex; height:5.676ex;" alt="{\displaystyle t=\tau t_{\text{c}}\Rightarrow dt=t_{\text{c}}d\tau \Rightarrow {\frac {d\tau }{dt}}={\frac {1}{t_{\text{c}}}}.}"></span> </p><p>The dimensionless differential operators with respect to the independent variable becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\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}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81c2beb0aa3bda3e774b138cadef1b0554efa439" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:65.602ex; height:6.176ex;" alt="{\displaystyle {\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}}}.}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Forcing_function">Forcing function</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=6" title="Edit section: Forcing function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If a system has a <a href="/wiki/Forcing_function_(differential_equations)" title="Forcing function (differential equations)">forcing function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=f(\tau t_{\text{c}})=f(t(\tau ))=F(\tau ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=f(\tau t_{\text{c}})=f(t(\tau ))=F(\tau ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/956dcf863b30be5988d667175dffd70af4a5b723" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.652ex; height:2.843ex;" alt="{\displaystyle f(t)=f(\tau t_{\text{c}})=f(t(\tau ))=F(\tau ).}"></span> </p><p>Hence, the new forcing function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> is made to be dependent on the dimensionless quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Linear_differential_equations_with_constant_coefficients">Linear differential equations with constant coefficients</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=7" title="Edit section: Linear differential equations with constant coefficients"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="First_order_system">First order system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=8" title="Edit section: First order system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the differential equation for a first order system: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\frac {dx}{dt}}+bx=Af(t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>=</mo> <mi>A</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\frac {dx}{dt}}+bx=Af(t).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b02f58af60f31b8d8e0c778b59aacc5619d6989" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.195ex; height:5.509ex;" alt="{\displaystyle a{\frac {dx}{dt}}+bx=Af(t).}"></span> </p><p>The derivation of the characteristic units to <b><a href="#math_1">Eq. 1</a></b> and <b><a href="#math_2">Eq. 2</a></b> for this system gave <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{\text{c}}={\frac {a}{b}},\ x_{\text{c}}={\frac {A}{b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mi>b</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{\text{c}}={\frac {a}{b}},\ x_{\text{c}}={\frac {A}{b}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c6ab46f337b49cd2ef4b6450458a7f6eff49f68" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.197ex; height:5.509ex;" alt="{\displaystyle t_{\text{c}}={\frac {a}{b}},\ x_{\text{c}}={\frac {A}{b}}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Second_order_system">Second order system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=9" title="Edit section: Second order system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A second order system has the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\frac {d^{2}x}{dt^{2}}}+b{\frac {dx}{dt}}+cx=Af(t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>c</mi> <mi>x</mi> <mo>=</mo> <mi>A</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\frac {d^{2}x}{dt^{2}}}+b{\frac {dx}{dt}}+cx=Af(t).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81403407f84ade41e70b7d59418cb8f3e223a980" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:27.48ex; height:6.009ex;" alt="{\displaystyle a{\frac {d^{2}x}{dt^{2}}}+b{\frac {dx}{dt}}+cx=Af(t).}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Substitution_step">Substitution step</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=10" title="Edit section: Substitution step"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Replace the variables <i>x</i> and <i>t</i> with their scaled quantities. The equation becomes </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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 }}+cx_{\text{c}}\chi =Af(\tau t_{\text{c}})=AF(\tau ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>χ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>χ</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>c</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mi>χ</mi> <mo>=</mo> <mi>A</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 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 }}+cx_{\text{c}}\chi =Af(\tau t_{\text{c}})=AF(\tau ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84da7d9c68fdcb1de0ed741f6e3f9005e58bbec3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.727ex; height:6.343ex;" alt="{\displaystyle 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 }}+cx_{\text{c}}\chi =Af(\tau t_{\text{c}})=AF(\tau ).}"></span> </p><p>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 </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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}}{ax_{\text{c}}}}F(\tau ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>χ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>χ</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>a</mi> </mfrac> </mrow> <mi>χ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>a</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\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}}{ax_{\text{c}}}}F(\tau ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6bfec8a9f11516a052e8e9a4535271fcb8338a7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:37.929ex; height:6.009ex;" alt="{\displaystyle {\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}}{ax_{\text{c}}}}F(\tau ).}"></span> </p><p>Now it is necessary to determine the quantities of <i>x</i><sub>c</sub> and <i>t</i><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. </p> <div class="mw-heading mw-heading4"><h4 id="Determination_of_characteristic_units">Determination of characteristic units</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=11" title="Edit section: Determination of characteristic units"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the variable <i>t</i><sub>c</sub>: </p> <ol><li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{\text{c}}={\frac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{\text{c}}={\frac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d57b3349482f60e346f162db2fbf97e5731f87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.966ex; height:4.843ex;" alt="{\displaystyle t_{\text{c}}={\frac {a}{b}}}"></span> the first order term is normalized.</li> <li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{\text{c}}={\sqrt {\frac {a}{c}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>a</mi> <mi>c</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{\text{c}}={\sqrt {\frac {a}{c}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6c16dd25894ee73b09300826c53bddf4bd4526f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.29ex; height:6.343ex;" alt="{\displaystyle t_{\text{c}}={\sqrt {\frac {a}{c}}}}"></span> the zeroth order term is normalized.</li></ol> <p>Both substitutions are valid. However, for pedagogical reasons, the latter substitution is used for second order systems. Choosing this substitution allows <i>x</i><sub>c</sub> to be determined by normalizing the coefficient of the forcing function: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1={\frac {At_{\text{c}}^{2}}{ax_{\text{c}}}}={\frac {A}{cx_{\text{c}}}}\Rightarrow x_{\text{c}}={\frac {A}{c}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mi>a</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mrow> <mi>c</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo stretchy="false">⇒</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mi>c</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1={\frac {At_{\text{c}}^{2}}{ax_{\text{c}}}}={\frac {A}{cx_{\text{c}}}}\Rightarrow x_{\text{c}}={\frac {A}{c}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31b3cd1d45badaa5ce60fd5da4789e4cd5658a15" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.198ex; height:5.843ex;" alt="{\displaystyle 1={\frac {At_{\text{c}}^{2}}{ax_{\text{c}}}}={\frac {A}{cx_{\text{c}}}}\Rightarrow x_{\text{c}}={\frac {A}{c}}.}"></span> </p><p>The differential equation becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}\chi }{d\tau ^{2}}}+{\frac {b}{\sqrt {ac}}}{\frac {d\chi }{d\tau }}+\chi =F(\tau ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>χ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <msqrt> <mi>a</mi> <mi>c</mi> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>χ</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>χ</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}\chi }{d\tau ^{2}}}+{\frac {b}{\sqrt {ac}}}{\frac {d\chi }{d\tau }}+\chi =F(\tau ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cde4a474e479b54fa10a5f5c795a730a6b121c0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:28.713ex; height:6.676ex;" alt="{\displaystyle {\frac {d^{2}\chi }{d\tau ^{2}}}+{\frac {b}{\sqrt {ac}}}{\frac {d\chi }{d\tau }}+\chi =F(\tau ).}"></span> </p><p>The coefficient of the first order term is unitless. Define <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\zeta \ {\stackrel {\mathrm {def} }{=}}\ {\frac {b}{\sqrt {ac}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>ζ</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <msqrt> <mi>a</mi> <mi>c</mi> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\zeta \ {\stackrel {\mathrm {def} }{=}}\ {\frac {b}{\sqrt {ac}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b72351db6e6b98f04d683c7bb70aad30a731041" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.33ex; height:6.343ex;" alt="{\displaystyle 2\zeta \ {\stackrel {\mathrm {def} }{=}}\ {\frac {b}{\sqrt {ac}}}.}"></span> </p><p>The factor 2 is present so that the solutions can be parameterized in terms of <i>ζ</i>. In the context of mechanical or electrical systems, <i>ζ</i> is known as the <a href="/wiki/Damping_ratio" class="mw-redirect" title="Damping ratio">damping ratio</a>, and is an important parameter required in the analysis of <a href="/wiki/Control_system" title="Control system">control systems</a>. 2<i>ζ</i> is also known as the <a href="/wiki/Linewidth" class="mw-redirect" title="Linewidth">linewidth</a> of the system. The result of the definition is the <a href="/wiki/Harmonic_oscillator#Universal_oscillator_equation" title="Harmonic oscillator">universal oscillator equation</a>. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}\chi }{d\tau ^{2}}}+2\zeta {\frac {d\chi }{d\tau }}+\chi =F(\tau ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>χ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>χ</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>χ</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}\chi }{d\tau ^{2}}}+2\zeta {\frac {d\chi }{d\tau }}+\chi =F(\tau ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e0fb21537d3c1a00c5f16e288adb0c045b131dd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.961ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}\chi }{d\tau ^{2}}}+2\zeta {\frac {d\chi }{d\tau }}+\chi =F(\tau ).}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Higher_order_systems">Higher order systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=12" title="Edit section: Higher order systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The general <i>n</i>th order linear differential equation with constant coefficients has the form: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 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).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d70bf3c0096d44bf3aeba5dd012bfea3dd424508" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:83.44ex; height:7.009ex;" alt="{\displaystyle 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).}"></span> </p><p>The function <i>f</i>(<i>t</i>) is known as the <a href="/wiki/Forcing_function_(differential_equations)" title="Forcing function (differential equations)">forcing function</a>. </p><p>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 <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">roots</a> of its <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a> are either <a href="/wiki/Real_number" title="Real number">real</a>, or <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> pairs. Therefore, understanding how nondimensionalization applies to first and second ordered systems allows the properties of higher order systems to be determined through <a href="/wiki/Superposition_principle" title="Superposition principle">superposition</a>. </p><p>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 <a href="/wiki/Symbolic_computation" class="mw-redirect" title="Symbolic computation">symbolic computation</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Examples_of_recovering_characteristic_units">Examples of recovering characteristic units</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=13" title="Edit section: Examples of recovering characteristic units"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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. </p> <div class="mw-heading mw-heading4"><h4 id="Mechanical_oscillations">Mechanical oscillations</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=14" title="Edit section: Mechanical oscillations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Mass-Spring-Damper.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Mass-Spring-Damper.png/300px-Mass-Spring-Damper.png" decoding="async" width="300" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Mass-Spring-Damper.png/450px-Mass-Spring-Damper.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Mass-Spring-Damper.png/600px-Mass-Spring-Damper.png 2x" data-file-width="763" data-file-height="448" /></a><figcaption>A mass attached to a spring and a damper.</figcaption></figure> <p>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 </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> = displacement from equilibrium [m]</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> = time [s]</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> = external force or "disturbance" applied to system [kg⋅m⋅s<sup>−2</sup>]</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> = mass of the block [kg]</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> = damping constant of dashpot [kg⋅s<sup>−1</sup>]</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> = force constant of spring [kg⋅s<sup>−2</sup>]</li></ul> <p>Suppose the applied force is a sinusoid <span class="nowrap"><i>F</i> = <i>F</i><sub>0</sub> cos(<i>ωt</i>)</span>, the differential equation that describes the motion of the block is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\frac {d^{2}x}{dt^{2}}}+B{\frac {dx}{dt}}+kx=F_{0}\cos(\omega t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>k</mi> <mi>x</mi> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\frac {d^{2}x}{dt^{2}}}+B{\frac {dx}{dt}}+kx=F_{0}\cos(\omega t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa4b807916887756e5943e708441b35d351c88e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:33.086ex; height:6.009ex;" alt="{\displaystyle m{\frac {d^{2}x}{dt^{2}}}+B{\frac {dx}{dt}}+kx=F_{0}\cos(\omega t)}"></span> </p><p>Nondimensionalizing this equation the same way as described under <a href="#Second_order_system">§ Second order system</a> yields several characteristics of the system: </p> <ul><li>The intrinsic unit <i>x</i><sub>c</sub> corresponds to the distance the block moves per unit force</li></ul> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{c}}={\frac {F_{0}}{k}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>k</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{c}}={\frac {F_{0}}{k}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c77c9c08a858dcd02050455b8474230ff11a3e60" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.422ex; height:5.509ex;" alt="{\displaystyle x_{\text{c}}={\frac {F_{0}}{k}}.}"></span> </p> <ul><li>The characteristic variable <i>t</i><sub>c</sub> is equal to the period of the oscillations</li></ul> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{\text{c}}={\sqrt {\frac {m}{k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>m</mi> <mi>k</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{\text{c}}={\sqrt {\frac {m}{k}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f094ddedcb481d29062a577ceebff3bcba3b024" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.101ex; height:6.176ex;" alt="{\displaystyle t_{\text{c}}={\sqrt {\frac {m}{k}}}}"></span> </p> <ul><li>The dimensionless variable 2<i>ζ</i> corresponds to the linewidth of the system.</li></ul> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\zeta ={\frac {B}{\sqrt {mk}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>ζ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>B</mi> <msqrt> <mi>m</mi> <mi>k</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\zeta ={\frac {B}{\sqrt {mk}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e090ff31a052e707b8474f37a5da6ece34b8b8ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.38ex; height:6.176ex;" alt="{\displaystyle 2\zeta ={\frac {B}{\sqrt {mk}}}}"></span> </p> <ul><li><i>ζ</i> itself is the <a href="/wiki/Damping_ratio" class="mw-redirect" title="Damping ratio">damping ratio</a></li></ul> <div class="mw-heading mw-heading4"><h4 id="Electrical_oscillations">Electrical oscillations</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=15" title="Edit section: Electrical oscillations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading5"><h5 id="First-order_series_RC_circuit">First-order series RC circuit</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=16" title="Edit section: First-order series RC circuit"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a series <a href="/wiki/RC_circuit" title="RC circuit">RC</a> attached to a <a href="/wiki/Power_supply" title="Power supply">voltage source</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R{\frac {dQ}{dt}}+{\frac {Q}{C}}=V(t)\Rightarrow {\frac {d\chi }{d\tau }}+\chi =F(\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>Q</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mi>C</mi> </mfrac> </mrow> <mo>=</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>χ</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>χ</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R{\frac {dQ}{dt}}+{\frac {Q}{C}}=V(t)\Rightarrow {\frac {d\chi }{d\tau }}+\chi =F(\tau )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26bb725d643e1514c0d8a8e04b180fbb1854c917" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.971ex; height:5.509ex;" alt="{\displaystyle R{\frac {dQ}{dt}}+{\frac {Q}{C}}=V(t)\Rightarrow {\frac {d\chi }{d\tau }}+\chi =F(\tau )}"></span> with substitutions <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\chi x_{\text{c}},\ t=\tau t_{\text{c}},\ x_{\text{c}}=CV_{0},\ t_{\text{c}}=RC,\ F=V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mi>χ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo>=</mo> <mi>τ</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>=</mo> <mi>C</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>=</mo> <mi>R</mi> <mi>C</mi> <mo>,</mo> <mtext> </mtext> <mi>F</mi> <mo>=</mo> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=\chi x_{\text{c}},\ t=\tau t_{\text{c}},\ x_{\text{c}}=CV_{0},\ t_{\text{c}}=RC,\ F=V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98b26ef84d569a686f264d96ad124fa3c63a60c5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:47.354ex; height:2.509ex;" alt="{\displaystyle Q=\chi x_{\text{c}},\ t=\tau t_{\text{c}},\ x_{\text{c}}=CV_{0},\ t_{\text{c}}=RC,\ F=V.}"></span> </p><p>The first characteristic unit corresponds to the total <a href="/wiki/Electric_charge" title="Electric charge">charge</a> in the circuit. The second characteristic unit corresponds to the <a href="/wiki/Time_constant" title="Time constant">time constant</a> for the system. </p> <div class="mw-heading mw-heading5"><h5 id="Second-order_series_RLC_circuit">Second-order series RLC circuit</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=17" title="Edit section: Second-order series RLC circuit"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a series configuration of <i>R</i>, <i>C</i>, <i>L</i> components where <i>Q</i> is the charge in the system <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L{\frac {d^{2}Q}{dt^{2}}}+R{\frac {dQ}{dt}}+{\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 )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>Q</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>Q</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mi>C</mi> </mfrac> </mrow> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>χ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>χ</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>χ</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L{\frac {d^{2}Q}{dt^{2}}}+R{\frac {dQ}{dt}}+{\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 )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50d6017cec84e83ef6eb849b3e77d6bf372b7811" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:65.617ex; height:6.009ex;" alt="{\displaystyle L{\frac {d^{2}Q}{dt^{2}}}+R{\frac {dQ}{dt}}+{\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 )}"></span> with the substitutions <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\chi x_{\text{c}},\ t=\tau t_{\text{c}},\ \ x_{\text{c}}=CV_{0},\ t_{\text{c}}={\sqrt {LC}},\ 2\zeta =R{\sqrt {\frac {C}{L}}},\ \Omega =t_{\text{c}}\omega .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mi>χ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo>=</mo> <mi>τ</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>=</mo> <mi>C</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>L</mi> <mi>C</mi> </msqrt> </mrow> <mo>,</mo> <mtext> </mtext> <mn>2</mn> <mi>ζ</mi> <mo>=</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>C</mi> <mi>L</mi> </mfrac> </msqrt> </mrow> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mi>ω</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=\chi x_{\text{c}},\ t=\tau t_{\text{c}},\ \ x_{\text{c}}=CV_{0},\ t_{\text{c}}={\sqrt {LC}},\ 2\zeta =R{\sqrt {\frac {C}{L}}},\ \Omega =t_{\text{c}}\omega .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81003e21c0258f933a8de067e24b52f69141546b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:64.748ex; height:6.176ex;" alt="{\displaystyle Q=\chi x_{\text{c}},\ t=\tau t_{\text{c}},\ \ x_{\text{c}}=CV_{0},\ t_{\text{c}}={\sqrt {LC}},\ 2\zeta =R{\sqrt {\frac {C}{L}}},\ \Omega =t_{\text{c}}\omega .}"></span> </p><p>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. </p> <div class="mw-heading mw-heading3"><h3 id="Quantum_mechanics">Quantum mechanics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=18" title="Edit section: Quantum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Quantum_harmonic_oscillator">Quantum harmonic oscillator</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=19" title="Edit section: Quantum harmonic oscillator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> for the one-dimensional time independent <a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillator</a> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+{\frac {1}{2}}m\omega ^{2}x^{2}\right)\psi (x)=E\psi (x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+{\frac {1}{2}}m\omega ^{2}x^{2}\right)\psi (x)=E\psi (x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a48082e148577da9031d29dc579194c4547dba24" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.68ex; height:6.343ex;" alt="{\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+{\frac {1}{2}}m\omega ^{2}x^{2}\right)\psi (x)=E\psi (x).}"></span> </p><p>The modulus square of the <a href="/wiki/Wavefunction" class="mw-redirect" title="Wavefunction">wavefunction</a> <span class="texhtml">|<i>ψ</i>(<i>x</i>)|<sup>2</sup></span> represents probability density that, when integrated over <span class="texhtml"><i>x</i></span>, gives a dimensionless probability. Therefore, <span class="texhtml">|<i>ψ</i>(<i>x</i>)|<sup>2</sup></span> has units of inverse length. To nondimensionalize this, it must be rewritten as a function of a dimensionless variable. To do this, we substitute <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {x}}\equiv {\frac {x}{x_{\text{c}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {x}}\equiv {\frac {x}{x_{\text{c}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e88022e055577f62735d0f23c15b1b1a5753144" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:8.203ex; height:5.009ex;" alt="{\displaystyle {\tilde {x}}\equiv {\frac {x}{x_{\text{c}}}},}"></span> where <span class="texhtml"><i>x</i><sub>c</sub></span> is some characteristic length of this system. This gives us a dimensionless wave function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\psi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\psi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4ac6d8bfc5b19aea7fd59edc84dca2a1fa3a86e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.596ex; height:3.009ex;" alt="{\displaystyle {\tilde {\psi }}}"></span> defined via <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)=\psi ({\tilde {x}}x_{\text{c}})=\psi (x(x_{\text{c}}))={\tilde {\psi }}({\tilde {x}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)=\psi ({\tilde {x}}x_{\text{c}})=\psi (x(x_{\text{c}}))={\tilde {\psi }}({\tilde {x}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29b11ac4a3323ad2f4d5fa81a310e70204af622f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.026ex; height:3.176ex;" alt="{\displaystyle \psi (x)=\psi ({\tilde {x}}x_{\text{c}})=\psi (x(x_{\text{c}}))={\tilde {\psi }}({\tilde {x}}).}"></span> </p><p>The differential equation then becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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 {2mx_{\text{c}}^{2}E}{\hbar ^{2}}}{\tilde {\psi }}({\tilde {x}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>E</mi> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \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 {2mx_{\text{c}}^{2}E}{\hbar ^{2}}}{\tilde {\psi }}({\tilde {x}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeffbf5b445433e5033ca10dc3a88754bc5ced81" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:95.308ex; height:6.343ex;" alt="{\displaystyle \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 {2mx_{\text{c}}^{2}E}{\hbar ^{2}}}{\tilde {\psi }}({\tilde {x}}).}"></span> </p><p>To make the term in front of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {x}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {x}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d6f2688d143e07bc22b34d0454976c60bc2a99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle {\tilde {x}}^{2}}"></span> dimensionless, set <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {m^{2}\omega ^{2}x_{\text{c}}^{4}}{\hbar ^{2}}}=1\Rightarrow x_{\text{c}}={\sqrt {\frac {\hbar }{m\omega }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo stretchy="false">⇒</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi class="MJX-variant">ℏ</mi> <mrow> <mi>m</mi> <mi>ω</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {m^{2}\omega ^{2}x_{\text{c}}^{4}}{\hbar ^{2}}}=1\Rightarrow x_{\text{c}}={\sqrt {\frac {\hbar }{m\omega }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2998a949b972aa718631001e36b349b3e6697fc4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.373ex; height:6.176ex;" alt="{\displaystyle {\frac {m^{2}\omega ^{2}x_{\text{c}}^{4}}{\hbar ^{2}}}=1\Rightarrow x_{\text{c}}={\sqrt {\frac {\hbar }{m\omega }}}.}"></span> </p><p>The fully nondimensionalized equation is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(-{\frac {d^{2}}{d{\tilde {x}}^{2}}}+{\tilde {x}}^{2}\right){\tilde {\psi }}({\tilde {x}})={\tilde {E}}{\tilde {\psi }}({\tilde {x}}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(-{\frac {d^{2}}{d{\tilde {x}}^{2}}}+{\tilde {x}}^{2}\right){\tilde {\psi }}({\tilde {x}})={\tilde {E}}{\tilde {\psi }}({\tilde {x}}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0857609d3a672729c9f8eeb8e90d1c95ebc959f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.267ex; height:6.343ex;" alt="{\displaystyle \left(-{\frac {d^{2}}{d{\tilde {x}}^{2}}}+{\tilde {x}}^{2}\right){\tilde {\psi }}({\tilde {x}})={\tilde {E}}{\tilde {\psi }}({\tilde {x}}),}"></span> where we have defined <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\equiv {\frac {\hbar \omega }{2}}{\tilde {E}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\equiv {\frac {\hbar \omega }{2}}{\tilde {E}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0efb3a523eaf40c323d4448258399207de1712e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.885ex; height:5.343ex;" alt="{\displaystyle E\equiv {\frac {\hbar \omega }{2}}{\tilde {E}}.}"></span> The factor in front of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18f629216bbaa99cf55e7cbf0f27fda050c95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.676ex;" alt="{\displaystyle {\tilde {E}}}"></span> is in fact (coincidentally) the <a href="/wiki/Ground_state" title="Ground state">ground state</a> energy of the harmonic oscillator. Usually, the energy term is not made dimensionless as we are interested in determining the energies of the <a href="/wiki/Quantum_state" title="Quantum state">quantum states</a>. Rearranging the first equation, the familiar equation for the harmonic oscillator becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\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}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a569080f98a4bcd74e56dbe1fa6dfce524e4f3e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.242ex; height:6.343ex;" alt="{\displaystyle {\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}}).}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Statistical_analogs">Statistical analogs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=20" title="Edit section: Statistical analogs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Normalization_(statistics)" title="Normalization (statistics)">Normalization (statistics)</a></div> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, the analogous process is usually dividing a difference (a distance) by a scale factor (a measure of <a href="/wiki/Statistical_dispersion" title="Statistical dispersion">statistical dispersion</a>), which yields a dimensionless number, which is called <i><a href="/wiki/Normalization_(statistics)" title="Normalization (statistics)">normalization</a>.</i> Most often, this is dividing <a href="/wiki/Errors_and_residuals_in_statistics" class="mw-redirect" title="Errors and residuals in statistics">errors or residuals</a> by the <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a> or sample standard deviation, respectively, yielding <a href="/wiki/Standard_score" title="Standard score">standard scores</a> and <a href="/wiki/Studentized_residual" title="Studentized residual">studentized residuals</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=21" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Buckingham_%CF%80_theorem" title="Buckingham π theorem">Buckingham π theorem</a></li> <li><a href="/wiki/Dimensionless_number" class="mw-redirect" title="Dimensionless number">Dimensionless number</a></li> <li><a href="/wiki/Natural_units" title="Natural units">Natural units</a></li> <li><a href="/wiki/System_equivalence" title="System equivalence">System equivalence</a></li> <li><a href="/wiki/RLC_circuit" title="RLC circuit">RLC circuit</a></li> <li><a href="/wiki/RL_circuit" title="RL circuit">RL circuit</a></li> <li><a href="/wiki/RC_circuit" title="RC circuit">RC circuit</a></li> <li><a href="/wiki/Logistic_map" title="Logistic map">Logistic equation</a></li> <li><a href="/wiki/Per-unit_system" title="Per-unit system">Per-unit system</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=22" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://scicomp.stackexchange.com/questions/41600/how-does-non-dimensionalization-improve-the-behavior-of-ode-solvers">"How does non-dimensionalization improve the behavior of ODE solvers?"</a>. <i>Computational Science Stack Exchange</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-08-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Computational+Science+Stack+Exchange&rft.atitle=How+does+non-dimensionalization+improve+the+behavior+of+ODE+solvers%3F&rft_id=https%3A%2F%2Fscicomp.stackexchange.com%2Fquestions%2F41600%2Fhow-does-non-dimensionalization-improve-the-behavior-of-ode-solvers&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANondimensionalization" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nondimensionalization&action=edit&section=23" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.royalsociety.org.nz/publications/journals/nzja/1998/059/">Analysis of differential equation models in biology: a case study for clover meristem populations</a> (Application of nondimensionalization to a problem in biology).</li> <li><a rel="nofollow" class="external text" href="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</a> <i>Jonathan Evans, Department of Mathematical Sciences, <a href="/wiki/University_of_Bath" title="University of Bath">University of Bath</a></i>. (see Chapter 3).</li> <li><a rel="nofollow" class="external text" href="https://hplgit.github.io/scaling-book/doc/pub/book/pdf/scaling-book-4screen-sol.pdf">Scaling of Differential Equations</a> <i>Hans Petter Langtangen, Geir K. Pedersen, Center for Biomedical Computing, Simula Research Laboratory and Department of Informatics, <a href="/wiki/University_of_Oslo" title="University of Oslo">University of Oslo</a></i>.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Nondimensionalization&oldid=1244787812">https://en.wikipedia.org/w/index.php?title=Nondimensionalization&oldid=1244787812</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Dimensional_analysis" title="Category:Dimensional analysis">Dimensional analysis</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 9 September 2024, at 05:36<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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