5598:
5131:
5593:{\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}}).}
1689:
3527:
1411:
2374:
3462:
4512:
2568:
4695:
6180:
1684:{\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}}}.}
648:
5916:
172:
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.
3517:
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
199:
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
2174:
3153:
4300:
2383:
5746:
4699:
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.
4869:
2815:
1404:
1014:
767:
3504:
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
4517:
6011:
2930:
5124:
494:
4261:
4125:
5753:
3137:
137:
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:
876:
3807:
3003:
1276:
1212:
5641:
2035:
4724:
2704:
1299:
2152:
5969:
1824:
655:
2369:{\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 ).}
4956:
489:
1092:
2822:
216:). 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.
4999:
3457:{\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).}
4507:{\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 )}
158:
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
3932:
3481:
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
2688:
908:
142:
3877:
2563:{\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 ).}
4130:
4018:
3983:
2378:
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
2643:
1949:
414:
3038:
5636:
4994:
4690:{\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 .}
3497:
pairs. Therefore, understanding how nondimensionalization applies to first and second ordered systems allows the properties of higher order systems to be determined through
5998:
6175:{\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}}).}
244:
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.
3692:
2937:
1288:
when nondimensionalizing. If differential operators are needed to describe the original system, their scaled counterparts become dimensionless differential operators.
1216:
1866:
1155:
789:
322:
296:
1728:
643:{\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 ).}
152:
3665:
3643:
3621:
3599:
3577:
3555:
1968:
1846:
147:
2694:
Both substitutions are valid. However, for pedagogical reasons, the latter substitution is used for second order systems. Choosing this substitution allows
5911:{\displaystyle \left(-{\frac {d^{2}}{d{\tilde {x}}^{2}}}+{\tilde {x}}^{2}\right){\tilde {\psi }}({\tilde {x}})={\tilde {E}}{\tilde {\psi }}({\tilde {x}}),}
3534:
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
2047:
807:
6297:
3475:
1699:
204:
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
5921:
4906:
has units of inverse length. To nondimensionalize this, it must be rewritten as a function of a dimensionless variable. To do this, we substitute
1733:
4909:
6004:
energy of the harmonic oscillator. Usually, the energy term is not made dimensionless as we are interested in determining the energies of the
1036:
237:
Choose judiciously the definition of the characteristic unit for each variable so that the coefficients of as many terms as possible become 1;
6330:
2586:
so that the coefficients become normalized. Since there are two free parameters, at most only two coefficients can be made to equal unity.
328:
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
3895:
6211:
3832:
84:
6346:
Hans Petter
Langtangen, Geir K. Pedersen, Center for Biomedical Computing, Simula Research Laboratory and Department of Informatics,
3947:
75:, in order to suggest that certain quantities are better measured relative to some appropriate unit. These units refer to quantities
5741:{\displaystyle {\frac {m^{2}\omega ^{2}x_{\text{c}}^{4}}{\hbar ^{2}}}=1\Rightarrow x_{\text{c}}={\sqrt {\frac {\hbar }{m\omega }}}.}
1884:
349:
4864:{\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).}
2810:{\displaystyle 1={\frac {At_{\text{c}}^{2}}{ax_{\text{c}}}}={\frac {A}{cx_{\text{c}}}}\Rightarrow x_{\text{c}}={\frac {A}{c}}.}
1399:{\displaystyle t=\tau t_{\text{c}}\Rightarrow dt=t_{\text{c}}d\tau \Rightarrow {\frac {d\tau }{dt}}={\frac {1}{t_{\text{c}}}}.}
433:
304:– represents the dependent variable – can be mass, voltage, or any measurable quantity. Its nondimensionalized counterpart is
260:". However, they are generally chosen so that it is convenient and intuitive to use for the problem at hand. For example, if "
1009:{\displaystyle {\frac {At_{\text{c}}}{ax_{\text{c}}}}={\frac {A}{bx_{\text{c}}}}=1\Rightarrow x_{\text{c}}={\frac {A}{b}}.}
6364:
102:, nondimensionalization can recover these values. The technique is especially useful for systems that can be described by
80:
2651:
762:{\displaystyle {\frac {d\chi }{d\tau }}+{\frac {bt_{\text{c}}}{a}}\chi ={\frac {At_{\text{c}}}{ax_{\text{c}}}}F(\tau ).}
90:
Nondimensionalization can also recover characteristic properties of a system. For example, if a system has an intrinsic
6342:
4718:
6205:
6190:
1130:
represent quantities with units. To scale these two variables, assume there are two intrinsic units of measurement
159:
3486:
2608:
46:
2925:{\displaystyle {\frac {d^{2}\chi }{d\tau ^{2}}}+{\frac {b}{\sqrt {ac}}}{\frac {d\chi }{d\tau }}+\chi =F(\tau ).}
87:
in an equation to intensive quantities, since the latter procedure results in variables that still carry units.
5119:{\displaystyle \psi (x)=\psi ({\tilde {x}}x_{\text{c}})=\psi (x(x_{\text{c}}))={\tilde {\psi }}({\tilde {x}}).}
652:
The coefficient of the highest ordered term is in front of the first derivative term. Dividing by this gives
6236:
4714:
3498:
1033:
The final dimensionless equation in this case becomes completely independent of any parameters with units:
6200:
1115:
231:
Replace each of them with a quantity scaled relative to a characteristic unit of measure to be determined;
4256:{\displaystyle Q=\chi x_{\text{c}},\ t=\tau t_{\text{c}},\ x_{\text{c}}=CV_{0},\ t_{\text{c}}=RC,\ F=V.}
4120:{\displaystyle R{\frac {dQ}{dt}}+{\frac {Q}{C}}=V(t)\Rightarrow {\frac {d\chi }{d\tau }}+\chi =F(\tau )}
343:
103:
38:
278:– represents the independent variable – usually a time quantity. Its nondimensionalized counterpart is
6324:
6199:, the analogous process is usually dividing a difference (a distance) by a scale factor (a measure of
5605:
4970:
1102:
Suppose for simplicity that a certain system is characterized by two variables – a dependent variable
6241:
6223:
3506:
192:. For such a system, it is advantageous to perform calculations relating to the swinging relative to
169:
58:
34:
3032:
50:
5974:
6347:
6335:
6325:
Analysis of differential equation models in biology: a case study for clover meristem populations
6251:
6215:
3482:
115:
3132:{\displaystyle {\frac {d^{2}\chi }{d\tau ^{2}}}+2\zeta {\frac {d\chi }{d\tau }}+\chi =F(\tau ).}
3494:
127:
123:
42:
6008:. Rearranging the first equation, the familiar equation for the harmonic oscillator becomes
4967:
is some characteristic length of this system. This gives us a dimensionless wave function
1851:
774:
307:
281:
6276:
4266:
1704:
1408:
The dimensionless differential operators with respect to the independent variable becomes
6219:
3650:
3628:
3606:
3584:
3562:
3540:
3020:
1831:
107:
6358:
6246:
6005:
4270:
3992:
3016:
234:
Divide through by the coefficient of the highest order polynomial or derivative term;
209:
111:
99:
871:{\displaystyle {\frac {bt_{\text{c}}}{a}}=1\Rightarrow t_{\text{c}}={\frac {a}{b}}.}
76:
6271:
6001:
4874:
4013:
17:
3802:{\displaystyle m{\frac {d^{2}x}{dt^{2}}}+B{\frac {dx}{dt}}+kx=F_{0}\cos(\omega t)}
3526:
2998:{\displaystyle 2\zeta \ {\stackrel {\mathrm {def} }{=}}\ {\frac {b}{\sqrt {ac}}}.}
196:. In some sense, this is normalizing the measurement with respect to the period.
6256:
3490:
54:
3150:
th order linear differential equation with constant coefficients has the form:
3007:
The factor 2 is present so that the solutions can be parameterized in terms of
1271:{\displaystyle \chi ={\frac {x}{x_{\text{c}}}}\Rightarrow x=\chi x_{\text{c}}.}
268:" might be an appropriate symbol to represent the dimensionless mass quantity.
240:
Rewrite the system of equations in terms of their new dimensionless quantities.
6266:
6261:
6196:
4009:
1207:{\displaystyle \tau ={\frac {t}{t_{\text{c}}}}\Rightarrow t=\tau t_{\text{c}}}
342:
As an illustrative example, consider a first order differential equation with
163:
3028:
186:
119:
91:
2030:{\displaystyle t_{\text{c}}={\frac {a}{b}},\ x_{\text{c}}={\frac {A}{b}}.}
2701:
to be determined by normalizing the coefficient of the forcing function:
182:
224:
To nondimensionalize a system of equations, one must do the following:
6298:"How does non-dimensionalization improve the behavior of ODE solvers?"
3689:, the differential equation that describes the motion of the block is
2147:{\displaystyle a{\frac {d^{2}x}{dt^{2}}}+b{\frac {dx}{dt}}+cx=Af(t).}
95:
62:
6331:
Course notes for
Mathematical Modelling and Industrial Mathematics
4269:
in the circuit. The second characteristic unit corresponds to the
3811:
Nondimensionalizing this equation the same way as described under
252:
There are no restrictions on the variable names used to replace "
6327:(Application of nondimensionalization to a problem in biology).
5964:{\displaystyle E\equiv {\frac {\hbar \omega }{2}}{\tilde {E}}.}
1819:{\displaystyle f(t)=f(\tau t_{\text{c}})=f(t(\tau ))=F(\tau ).}
4951:{\displaystyle {\tilde {x}}\equiv {\frac {x}{x_{\text{c}}}},}
1881:
Consider the differential equation for a first order system:
3019:, and is an important parameter required in the analysis of
2934:
The coefficient of the first order term is unitless. Define
798:, hence it is easiest to choose to set this to unity first:
271:
In this article, the following conventions have been used:
143:
List of dynamical systems and differential equations topics
83:
units. Nondimensionalization is not the same as converting
4888:
represents probability density that, when integrated over
3826:
corresponds to the distance the block moves per unit force
126:; a more natural pair of characteristic units is mean age/
27:
Mathematical simplification technique in physical sciences
484:{\displaystyle x=\chi x_{\text{c}},\ t=\tau t_{\text{c}}}
6203:), which yields a dimensionless number, which is called
1872:
Linear differential equations with constant coefficients
1087:{\displaystyle {\frac {d\chi }{d\tau }}+\chi =F(\tau ).}
4265:
The first characteristic unit corresponds to the total
3011:. In the context of mechanical or electrical systems,
1848:
is made to be dependent on the dimensionless quantity
6334:
6218:
or sample standard deviation, respectively, yielding
6014:
5977:
5924:
5756:
5644:
5608:
5134:
5002:
4973:
4912:
4727:
4520:
4303:
4133:
4021:
3950:
3898:
3835:
3695:
3653:
3631:
3609:
3587:
3565:
3543:
3156:
3041:
2940:
2825:
2707:
2654:
2611:
2386:
2177:
2050:
1971:
1887:
1854:
1834:
1736:
1707:
1414:
1302:
1219:
1158:
1039:
911:
810:
777:
658:
497:
436:
352:
310:
284:
228:
Identify all the independent and dependent variables;
3601:= external force or "disturbance" applied to system
3927:{\displaystyle t_{\text{c}}={\sqrt {\frac {m}{k}}}}
3031:of the system. The result of the definition is the
2683:{\displaystyle t_{\text{c}}={\sqrt {\frac {a}{c}}}}
2572:Now it is necessary to determine the quantities of
2169:with their scaled quantities. The equation becomes
6174:
5992:
5963:
5910:
5740:
5630:
5592:
5118:
4988:
4950:
4863:
4689:
4506:
4255:
4119:
3977:
3926:
3871:
3801:
3659:
3637:
3615:
3593:
3571:
3549:
3456:
3131:
2997:
2924:
2809:
2682:
2637:
2562:
2368:
2146:
2029:
1943:
1860:
1840:
1818:
1722:
1683:
1398:
1270:
1206:
1086:
1008:
870:
783:
761:
642:
483:
419:In this equation the independent variable here is
408:
316:
290:
110:. One of the simplest characteristic units is the
4894:, gives a dimensionless probability. Therefore,
3872:{\displaystyle x_{\text{c}}={\frac {F_{0}}{k}}.}
3978:{\displaystyle 2\zeta ={\frac {B}{\sqrt {mk}}}}
1152:respectively, such that these conditions hold:
3815:yields several characteristics of the system:
1953:The derivation of the characteristic units to
153:Differential equations of mathematical physics
3410:
3387:
339:is the characteristic unit used to scale it.
57:units are involved. It is closely related to
8:
148:List of partial differential equation topics
3941:corresponds to the linewidth of the system.
3513:Examples of recovering characteristic units
2638:{\displaystyle t_{\text{c}}={\frac {a}{b}}}
1944:{\displaystyle a{\frac {dx}{dt}}+bx=Af(t).}
409:{\displaystyle a{\frac {dx}{dt}}+bx=Af(t).}
3889:is equal to the period of the oscillations
791:only contains one characteristic variable
106:. One important use is in the analysis of
6155:
6154:
6140:
6139:
6119:
6118:
6104:
6103:
6092:
6081:
6080:
6067:
6056:
6055:
6044:
6038:
6015:
6013:
5979:
5978:
5976:
5947:
5946:
5931:
5923:
5891:
5890:
5876:
5875:
5864:
5863:
5846:
5845:
5831:
5830:
5819:
5808:
5807:
5794:
5783:
5782:
5771:
5765:
5755:
5750:The fully nondimensionalized equation is
5718:
5709:
5688:
5677:
5672:
5662:
5652:
5645:
5643:
5622:
5611:
5610:
5607:
5573:
5572:
5558:
5557:
5549:
5535:
5530:
5517:
5500:
5499:
5485:
5484:
5473:
5462:
5461:
5452:
5441:
5436:
5426:
5416:
5409:
5397:
5386:
5385:
5374:
5368:
5343:
5342:
5328:
5327:
5326:
5306:
5305:
5291:
5290:
5279:
5268:
5267:
5260:
5255:
5245:
5228:
5216:
5205:
5204:
5193:
5187:
5179:
5174:
5165:
5149:
5143:
5133:
5099:
5098:
5084:
5083:
5068:
5040:
5025:
5024:
5001:
4975:
4974:
4972:
4937:
4928:
4914:
4913:
4911:
4817:
4807:
4790:
4778:
4764:
4758:
4742:
4736:
4726:
4717:for the one-dimensional time independent
4675:
4647:
4619:
4610:
4594:
4578:
4559:
4534:
4519:
4457:
4439:
4421:
4414:
4387:
4370:
4347:
4332:
4314:
4307:
4302:
4220:
4204:
4188:
4172:
4147:
4132:
4076:
4048:
4025:
4020:
3960:
3949:
3912:
3903:
3897:
3855:
3849:
3840:
3834:
3775:
3739:
3724:
3706:
3699:
3694:
3652:
3630:
3608:
3586:
3564:
3542:
3530:A mass attached to a spring and a damper.
3415:
3409:
3408:
3392:
3386:
3385:
3379:
3369:
3358:
3333:
3299:
3293:
3253:
3233:
3227:
3215:
3187:
3173:
3167:
3161:
3155:
3085:
3067:
3049:
3042:
3040:
2977:
2959:
2958:
2953:
2951:
2950:
2939:
2878:
2863:
2851:
2833:
2826:
2824:
2794:
2785:
2769:
2756:
2744:
2729:
2724:
2714:
2706:
2668:
2659:
2653:
2625:
2616:
2610:
2536:
2521:
2514:
2509:
2502:
2486:
2480:
2473:
2468:
2444:
2434:
2428:
2412:
2394:
2387:
2385:
2336:
2308:
2278:
2270:
2260:
2254:
2239:
2221:
2214:
2206:
2199:
2194:
2187:
2181:
2176:
2094:
2079:
2061:
2054:
2049:
2014:
2005:
1985:
1976:
1970:
1891:
1886:
1853:
1833:
1765:
1735:
1706:
1669:
1655:
1649:
1641:
1634:
1629:
1623:
1614:
1594:
1586:
1577:
1562:
1543:
1526:
1512:
1506:
1488:
1480:
1471:
1453:
1433:
1415:
1413:
1385:
1376:
1353:
1338:
1316:
1301:
1259:
1235:
1226:
1218:
1198:
1174:
1165:
1157:
1040:
1038:
993:
984:
962:
949:
937:
922:
912:
910:
855:
846:
821:
811:
809:
776:
735:
720:
710:
692:
682:
659:
657:
604:
603:
598:
596:
595:
583:
555:
525:
517:
507:
501:
496:
475:
450:
435:
356:
351:
309:
283:
79:to the system, rather than units such as
3812:
3671:Suppose the applied force is a sinusoid
3525:
6289:
6018:
5934:
5721:
5685:
5546:
5449:
5146:
5128:The differential equation then becomes
4739:
2590:Determination of characteristic units
7:
6302:Computational Science Stack Exchange
2690:the zeroth order term is normalized.
1280:These equations are used to replace
902:
801:
2645:the first order term is normalized.
2044:A second order system has the form
4665:
4495:
2966:
2963:
2960:
2819:The differential equation becomes
611:
608:
605:
49:. This technique can simplify and
33:is the partial or full removal of
25:
6343:Scaling of Differential Equations
6000:is in fact (coincidentally) the
5631:{\displaystyle {\tilde {x}}^{2}}
4989:{\displaystyle {\tilde {\psi }}}
3557:= displacement from equilibrium
1828:Hence, the new forcing function
423:, and the dependent variable is
264:" represented mass, the letter "
4277:Second-order series RLC circuit
491:. This results in the equation
6166:
6160:
6151:
6145:
6130:
6124:
6115:
6109:
6086:
6061:
5984:
5952:
5902:
5896:
5887:
5881:
5869:
5857:
5851:
5842:
5836:
5813:
5788:
5702:
5616:
5584:
5578:
5569:
5563:
5511:
5505:
5496:
5490:
5467:
5391:
5357:
5354:
5348:
5339:
5333:
5317:
5311:
5302:
5296:
5273:
5210:
5110:
5104:
5095:
5089:
5077:
5074:
5061:
5055:
5046:
5030:
5021:
5012:
5006:
4980:
4919:
4855:
4849:
4837:
4831:
4501:
4492:
4411:
4408:
4399:
4281:For a series configuration of
4114:
4108:
4073:
4070:
4064:
3796:
3787:
3645:= damping constant of dashpot
3448:
3442:
3430:
3424:
3348:
3342:
3323:
3317:
3277:
3271:
3205:
3199:
3123:
3117:
2916:
2910:
2778:
2554:
2548:
2360:
2354:
2342:
2326:
2138:
2132:
1935:
1929:
1810:
1804:
1795:
1792:
1786:
1780:
1771:
1755:
1746:
1740:
1717:
1711:
1503:
1350:
1322:
1243:
1182:
1078:
1072:
977:
839:
753:
747:
634:
628:
589:
573:
400:
394:
185:is swinging with a particular
1:
6210:Most often, this is dividing
5602:To make the term in front of
4004:First-order series RC circuit
3033:universal oscillator equation
71:is used interchangeably with
5993:{\displaystyle {\tilde {E}}}
4297:is the charge in the system
3937:The dimensionless variable 2
3882:The characteristic variable
1106:and an independent variable
771:The coefficient in front of
4719:quantum harmonic oscillator
4709:Quantum harmonic oscillator
3667:= force constant of spring
1961:
1955:
220:Nondimensionalization steps
130:, which correspond to base
6381:
6191:Normalization (statistics)
6188:
4873:The modulus square of the
3813:§ Second order system
1296:Consider the relationship
3487:characteristic polynomial
122:of a system experiencing
114:of a system experiencing
47:substitution of variables
5971:The factor in front of
4514:with the substitutions
3999:Electrical oscillations
3522:Mechanical oscillations
1144:with the same units as
6201:statistical dispersion
6176:
5994:
5965:
5918:where we have defined
5912:
5742:
5632:
5594:
5120:
4990:
4952:
4865:
4691:
4508:
4257:
4121:
3979:
3928:
3873:
3803:
3661:
3639:
3617:
3595:
3573:
3551:
3531:
3458:
3374:
3133:
2999:
2926:
2811:
2684:
2639:
2594:Consider the variable
2564:
2370:
2161:Replace the variables
2148:
2031:
1945:
1862:
1842:
1820:
1724:
1685:
1400:
1292:Differential operators
1272:
1208:
1088:
1010:
872:
785:
763:
644:
485:
410:
318:
292:
104:differential equations
6224:studentized residuals
6177:
5995:
5966:
5913:
5743:
5633:
5595:
5121:
4991:
4953:
4866:
4692:
4509:
4258:
4122:
3980:
3929:
3874:
3804:
3662:
3640:
3618:
3596:
3574:
3552:
3529:
3459:
3354:
3134:
3027:is also known as the
3000:
2927:
2812:
2685:
2640:
2565:
2371:
2149:
2032:
1965:for this system gave
1946:
1863:
1861:{\displaystyle \tau }
1843:
1821:
1725:
1686:
1401:
1273:
1209:
1089:
1011:
873:
786:
784:{\displaystyle \chi }
764:
645:
486:
411:
344:constant coefficients
319:
317:{\displaystyle \chi }
293:
291:{\displaystyle \tau }
73:nondimensionalization
31:Nondimensionalization
6365:Dimensional analysis
6242:Dimensionless number
6237:Buckingham π theorem
6012:
5975:
5922:
5754:
5642:
5606:
5132:
5000:
4971:
4910:
4725:
4715:Schrödinger equation
4518:
4301:
4131:
4019:
3948:
3896:
3833:
3693:
3651:
3629:
3623:= mass of the block
3607:
3585:
3563:
3541:
3507:symbolic computation
3154:
3142:Higher order systems
3039:
2938:
2823:
2705:
2652:
2609:
2384:
2175:
2048:
1969:
1885:
1852:
1832:
1734:
1723:{\displaystyle f(t)}
1705:
1412:
1300:
1217:
1156:
1037:
909:
808:
775:
656:
495:
434:
350:
332:is a quantity, then
308:
282:
202:characteristic units
134:rather than base 2.
118:, or conversely the
85:extensive quantities
59:dimensional analysis
6212:errors or residuals
6185:Statistical analogs
5682:
5638:dimensionless, set
5540:
5446:
5265:
5184:
4127:with substitutions
3819:The intrinsic unit
2734:
2040:Second order system
92:resonance frequency
61:. In some physical
43:physical quantities
35:physical dimensions
18:Characteristic unit
6348:University of Oslo
6339:. (see Chapter 3).
6336:University of Bath
6252:System equivalence
6216:standard deviation
6172:
5990:
5961:
5908:
5738:
5668:
5628:
5590:
5526:
5432:
5251:
5170:
5116:
4986:
4948:
4861:
4687:
4504:
4253:
4117:
3975:
3924:
3869:
3799:
3657:
3635:
3613:
3591:
3569:
3547:
3532:
3474:) is known as the
3454:
3129:
2995:
2922:
2807:
2720:
2680:
2635:
2560:
2366:
2144:
2027:
1941:
1877:First order system
1858:
1838:
1816:
1720:
1698:If a system has a
1681:
1396:
1268:
1204:
1084:
1006:
868:
781:
759:
640:
481:
406:
314:
288:
116:exponential growth
6272:Logistic equation
6163:
6148:
6127:
6112:
6089:
6074:
6064:
6028:
5987:
5955:
5944:
5899:
5884:
5872:
5854:
5839:
5816:
5801:
5791:
5733:
5732:
5712:
5694:
5675:
5619:
5581:
5566:
5555:
5533:
5508:
5493:
5470:
5458:
5439:
5404:
5394:
5351:
5336:
5314:
5299:
5276:
5258:
5236:
5223:
5213:
5185:
5177:
5163:
5107:
5092:
5071:
5043:
5033:
4983:
4943:
4940:
4922:
4798:
4785:
4756:
4704:Quantum mechanics
4678:
4664:
4657:
4656:
4634:
4627:
4613:
4605:
4581:
4573:
4570:
4562:
4545:
4537:
4475:
4446:
4378:
4365:
4339:
4293:components where
4240:
4223:
4215:
4191:
4183:
4175:
4158:
4150:
4094:
4056:
4043:
3973:
3972:
3922:
3921:
3906:
3864:
3843:
3757:
3731:
3660:{\displaystyle k}
3638:{\displaystyle B}
3616:{\displaystyle m}
3594:{\displaystyle f}
3572:{\displaystyle t}
3550:{\displaystyle x}
3495:complex conjugate
3405:
3312:
3266:
3194:
3103:
3074:
2990:
2989:
2976:
2971:
2949:
2896:
2876:
2875:
2858:
2802:
2788:
2776:
2772:
2751:
2747:
2727:
2678:
2677:
2662:
2633:
2619:
2543:
2539:
2517:
2494:
2476:
2462:
2442:
2431:
2419:
2339:
2311:
2296:
2276:
2273:
2263:
2246:
2212:
2202:
2190:
2157:Substitution step
2112:
2086:
2022:
2008:
2000:
1993:
1979:
1909:
1841:{\displaystyle F}
1768:
1676:
1647:
1637:
1607:
1592:
1589:
1556:
1533:
1501:
1486:
1483:
1466:
1451:
1428:
1391:
1388:
1371:
1341:
1319:
1262:
1241:
1238:
1201:
1180:
1177:
1058:
1030:
1029:
1001:
987:
969:
965:
944:
940:
925:
892:
891:
863:
849:
831:
824:
742:
738:
723:
702:
695:
677:
621:
616:
594:
586:
558:
543:
523:
520:
510:
478:
461:
453:
374:
170:Measuring devices
124:exponential decay
16:(Redirected from
6372:
6312:
6311:
6309:
6308:
6294:
6181:
6179:
6178:
6173:
6165:
6164:
6156:
6150:
6149:
6141:
6129:
6128:
6120:
6114:
6113:
6105:
6102:
6098:
6097:
6096:
6091:
6090:
6082:
6075:
6073:
6072:
6071:
6066:
6065:
6057:
6049:
6048:
6039:
6029:
6024:
6016:
5999:
5997:
5996:
5991:
5989:
5988:
5980:
5970:
5968:
5967:
5962:
5957:
5956:
5948:
5945:
5940:
5932:
5917:
5915:
5914:
5909:
5901:
5900:
5892:
5886:
5885:
5877:
5874:
5873:
5865:
5856:
5855:
5847:
5841:
5840:
5832:
5829:
5825:
5824:
5823:
5818:
5817:
5809:
5802:
5800:
5799:
5798:
5793:
5792:
5784:
5776:
5775:
5766:
5747:
5745:
5744:
5739:
5734:
5731:
5720:
5719:
5714:
5713:
5710:
5695:
5693:
5692:
5683:
5681:
5676:
5673:
5667:
5666:
5657:
5656:
5646:
5637:
5635:
5634:
5629:
5627:
5626:
5621:
5620:
5612:
5599:
5597:
5596:
5591:
5583:
5582:
5574:
5568:
5567:
5559:
5556:
5554:
5553:
5544:
5539:
5534:
5531:
5518:
5510:
5509:
5501:
5495:
5494:
5486:
5483:
5479:
5478:
5477:
5472:
5471:
5463:
5459:
5457:
5456:
5447:
5445:
5440:
5437:
5431:
5430:
5421:
5420:
5410:
5405:
5403:
5402:
5401:
5396:
5395:
5387:
5379:
5378:
5369:
5353:
5352:
5344:
5338:
5337:
5329:
5316:
5315:
5307:
5301:
5300:
5292:
5289:
5285:
5284:
5283:
5278:
5277:
5269:
5264:
5259:
5256:
5250:
5249:
5237:
5229:
5224:
5222:
5221:
5220:
5215:
5214:
5206:
5198:
5197:
5188:
5186:
5183:
5178:
5175:
5166:
5164:
5162:
5154:
5153:
5144:
5125:
5123:
5122:
5117:
5109:
5108:
5100:
5094:
5093:
5085:
5073:
5072:
5069:
5045:
5044:
5041:
5035:
5034:
5026:
4995:
4993:
4992:
4987:
4985:
4984:
4976:
4966:
4957:
4955:
4954:
4949:
4944:
4942:
4941:
4938:
4929:
4924:
4923:
4915:
4905:
4893:
4887:
4870:
4868:
4867:
4862:
4827:
4823:
4822:
4821:
4812:
4811:
4799:
4791:
4786:
4784:
4783:
4782:
4769:
4768:
4759:
4757:
4755:
4747:
4746:
4737:
4696:
4694:
4693:
4688:
4680:
4679:
4676:
4662:
4658:
4649:
4648:
4632:
4628:
4620:
4615:
4614:
4611:
4603:
4599:
4598:
4583:
4582:
4579:
4571:
4568:
4564:
4563:
4560:
4543:
4539:
4538:
4535:
4513:
4511:
4510:
4505:
4476:
4474:
4466:
4458:
4447:
4445:
4444:
4443:
4430:
4426:
4425:
4415:
4392:
4391:
4379:
4371:
4366:
4364:
4356:
4348:
4340:
4338:
4337:
4336:
4323:
4319:
4318:
4308:
4273:for the system.
4262:
4260:
4259:
4254:
4238:
4225:
4224:
4221:
4213:
4209:
4208:
4193:
4192:
4189:
4181:
4177:
4176:
4173:
4156:
4152:
4151:
4148:
4126:
4124:
4123:
4118:
4095:
4093:
4085:
4077:
4057:
4049:
4044:
4042:
4034:
4026:
3984:
3982:
3981:
3976:
3974:
3965:
3961:
3933:
3931:
3930:
3925:
3923:
3914:
3913:
3908:
3907:
3904:
3878:
3876:
3875:
3870:
3865:
3860:
3859:
3850:
3845:
3844:
3841:
3808:
3806:
3805:
3800:
3780:
3779:
3758:
3756:
3748:
3740:
3732:
3730:
3729:
3728:
3715:
3711:
3710:
3700:
3688:
3666:
3664:
3663:
3658:
3644:
3642:
3641:
3636:
3622:
3620:
3619:
3614:
3600:
3598:
3597:
3592:
3578:
3576:
3575:
3570:
3556:
3554:
3553:
3548:
3476:forcing function
3463:
3461:
3460:
3455:
3420:
3419:
3414:
3413:
3406:
3404:
3393:
3391:
3390:
3384:
3383:
3373:
3368:
3338:
3337:
3313:
3311:
3300:
3298:
3297:
3267:
3265:
3264:
3263:
3244:
3243:
3228:
3226:
3225:
3195:
3193:
3192:
3191:
3178:
3177:
3168:
3166:
3165:
3138:
3136:
3135:
3130:
3104:
3102:
3094:
3086:
3075:
3073:
3072:
3071:
3058:
3054:
3053:
3043:
3015:is known as the
3004:
3002:
3001:
2996:
2991:
2982:
2978:
2974:
2973:
2972:
2970:
2969:
2957:
2952:
2947:
2931:
2929:
2928:
2923:
2897:
2895:
2887:
2879:
2877:
2868:
2864:
2859:
2857:
2856:
2855:
2842:
2838:
2837:
2827:
2816:
2814:
2813:
2808:
2803:
2795:
2790:
2789:
2786:
2777:
2775:
2774:
2773:
2770:
2757:
2752:
2750:
2749:
2748:
2745:
2735:
2733:
2728:
2725:
2715:
2689:
2687:
2686:
2681:
2679:
2670:
2669:
2664:
2663:
2660:
2644:
2642:
2641:
2636:
2634:
2626:
2621:
2620:
2617:
2569:
2567:
2566:
2561:
2544:
2542:
2541:
2540:
2537:
2527:
2526:
2525:
2520:
2519:
2518:
2515:
2503:
2495:
2487:
2485:
2484:
2479:
2478:
2477:
2474:
2463:
2461:
2453:
2445:
2443:
2435:
2433:
2432:
2429:
2420:
2418:
2417:
2416:
2403:
2399:
2398:
2388:
2375:
2373:
2372:
2367:
2341:
2340:
2337:
2313:
2312:
2309:
2297:
2295:
2287:
2279:
2277:
2275:
2274:
2271:
2265:
2264:
2261:
2255:
2247:
2245:
2244:
2243:
2230:
2226:
2225:
2215:
2213:
2211:
2210:
2205:
2204:
2203:
2200:
2192:
2191:
2188:
2182:
2153:
2151:
2150:
2145:
2113:
2111:
2103:
2095:
2087:
2085:
2084:
2083:
2070:
2066:
2065:
2055:
2036:
2034:
2033:
2028:
2023:
2015:
2010:
2009:
2006:
1998:
1994:
1986:
1981:
1980:
1977:
1950:
1948:
1947:
1942:
1910:
1908:
1900:
1892:
1867:
1865:
1864:
1859:
1847:
1845:
1844:
1839:
1825:
1823:
1822:
1817:
1770:
1769:
1766:
1729:
1727:
1726:
1721:
1700:forcing function
1694:Forcing function
1690:
1688:
1687:
1682:
1677:
1675:
1674:
1673:
1660:
1659:
1650:
1648:
1646:
1645:
1640:
1639:
1638:
1635:
1624:
1619:
1618:
1613:
1609:
1608:
1606:
1595:
1593:
1591:
1590:
1587:
1578:
1567:
1566:
1561:
1557:
1555:
1544:
1534:
1532:
1531:
1530:
1517:
1516:
1507:
1502:
1500:
1489:
1487:
1485:
1484:
1481:
1472:
1467:
1465:
1454:
1452:
1450:
1442:
1434:
1429:
1427:
1416:
1405:
1403:
1402:
1397:
1392:
1390:
1389:
1386:
1377:
1372:
1370:
1362:
1354:
1343:
1342:
1339:
1321:
1320:
1317:
1277:
1275:
1274:
1269:
1264:
1263:
1260:
1242:
1240:
1239:
1236:
1227:
1213:
1211:
1210:
1205:
1203:
1202:
1199:
1181:
1179:
1178:
1175:
1166:
1093:
1091:
1090:
1085:
1059:
1057:
1049:
1041:
1024:
1015:
1013:
1012:
1007:
1002:
994:
989:
988:
985:
970:
968:
967:
966:
963:
950:
945:
943:
942:
941:
938:
928:
927:
926:
923:
913:
903:
886:
877:
875:
874:
869:
864:
856:
851:
850:
847:
832:
827:
826:
825:
822:
812:
802:
790:
788:
787:
782:
768:
766:
765:
760:
743:
741:
740:
739:
736:
726:
725:
724:
721:
711:
703:
698:
697:
696:
693:
683:
678:
676:
668:
660:
649:
647:
646:
641:
619:
618:
617:
615:
614:
602:
597:
592:
588:
587:
584:
560:
559:
556:
544:
542:
534:
526:
524:
522:
521:
518:
512:
511:
508:
502:
490:
488:
487:
482:
480:
479:
476:
459:
455:
454:
451:
415:
413:
412:
407:
375:
373:
365:
357:
323:
321:
320:
315:
297:
295:
294:
289:
21:
6380:
6379:
6375:
6374:
6373:
6371:
6370:
6369:
6355:
6354:
6321:
6316:
6315:
6306:
6304:
6296:
6295:
6291:
6286:
6281:
6277:Per-unit system
6232:
6220:standard scores
6193:
6187:
6079:
6054:
6050:
6040:
6034:
6030:
6017:
6010:
6009:
5973:
5972:
5933:
5920:
5919:
5806:
5781:
5777:
5767:
5761:
5757:
5752:
5751:
5724:
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5684:
5658:
5648:
5647:
5640:
5639:
5609:
5604:
5603:
5545:
5519:
5460:
5448:
5422:
5412:
5411:
5384:
5380:
5370:
5364:
5360:
5266:
5241:
5203:
5199:
5189:
5155:
5145:
5139:
5135:
5130:
5129:
5064:
5036:
4998:
4997:
4969:
4968:
4965:
4959:
4933:
4908:
4907:
4895:
4889:
4877:
4813:
4803:
4774:
4770:
4760:
4748:
4738:
4732:
4728:
4723:
4722:
4711:
4706:
4671:
4606:
4590:
4574:
4555:
4530:
4516:
4515:
4467:
4459:
4435:
4431:
4417:
4416:
4383:
4357:
4349:
4328:
4324:
4310:
4309:
4299:
4298:
4279:
4216:
4200:
4184:
4168:
4143:
4129:
4128:
4086:
4078:
4035:
4027:
4017:
4016:
4006:
4001:
3946:
3945:
3899:
3894:
3893:
3888:
3851:
3836:
3831:
3830:
3825:
3771:
3749:
3741:
3720:
3716:
3702:
3701:
3691:
3690:
3682:
3672:
3649:
3648:
3627:
3626:
3605:
3604:
3583:
3582:
3561:
3560:
3539:
3538:
3524:
3515:
3407:
3397:
3375:
3329:
3304:
3289:
3249:
3245:
3229:
3211:
3183:
3179:
3169:
3157:
3152:
3151:
3144:
3095:
3087:
3063:
3059:
3045:
3044:
3037:
3036:
3021:control systems
2936:
2935:
2888:
2880:
2847:
2843:
2829:
2828:
2821:
2820:
2781:
2765:
2761:
2740:
2736:
2716:
2703:
2702:
2700:
2655:
2650:
2649:
2612:
2607:
2606:
2600:
2592:
2585:
2578:
2532:
2528:
2510:
2508:
2504:
2469:
2467:
2454:
2446:
2424:
2408:
2404:
2390:
2389:
2382:
2381:
2332:
2304:
2288:
2280:
2266:
2256:
2235:
2231:
2217:
2216:
2195:
2193:
2183:
2173:
2172:
2159:
2104:
2096:
2075:
2071:
2057:
2056:
2046:
2045:
2042:
2001:
1972:
1967:
1966:
1901:
1893:
1883:
1882:
1879:
1874:
1850:
1849:
1830:
1829:
1761:
1732:
1731:
1703:
1702:
1696:
1665:
1661:
1651:
1630:
1628:
1599:
1582:
1576:
1572:
1571:
1548:
1539:
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1522:
1518:
1508:
1493:
1476:
1458:
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1420:
1410:
1409:
1381:
1363:
1355:
1334:
1312:
1298:
1297:
1294:
1255:
1231:
1215:
1214:
1194:
1170:
1154:
1153:
1143:
1136:
1100:
1050:
1042:
1035:
1034:
1022:
980:
958:
954:
933:
929:
918:
914:
907:
906:
884:
842:
817:
813:
806:
805:
797:
773:
772:
731:
727:
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712:
688:
684:
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579:
551:
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492:
471:
446:
432:
431:
366:
358:
348:
347:
338:
306:
305:
280:
279:
250:
222:
179:
108:control systems
53:problems where
28:
23:
22:
15:
12:
11:
5:
6378:
6376:
6368:
6367:
6357:
6356:
6353:
6352:
6340:
6328:
6320:
6319:External links
6317:
6314:
6313:
6288:
6287:
6285:
6282:
6280:
6279:
6274:
6269:
6264:
6259:
6254:
6249:
6244:
6239:
6233:
6231:
6228:
6189:Main article:
6186:
6183:
6171:
6168:
6162:
6159:
6153:
6147:
6144:
6138:
6135:
6132:
6126:
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6078:
6070:
6063:
6060:
6053:
6047:
6043:
6037:
6033:
6027:
6023:
6020:
6006:quantum states
5986:
5983:
5960:
5954:
5951:
5943:
5939:
5936:
5930:
5927:
5907:
5904:
5898:
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5889:
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5871:
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5862:
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5853:
5850:
5844:
5838:
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5805:
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5774:
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5482:
5476:
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5451:
5444:
5435:
5429:
5425:
5419:
5415:
5408:
5400:
5393:
5390:
5383:
5377:
5373:
5367:
5363:
5359:
5356:
5350:
5347:
5341:
5335:
5332:
5325:
5322:
5319:
5313:
5310:
5304:
5298:
5295:
5288:
5282:
5275:
5272:
5263:
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5248:
5244:
5240:
5235:
5232:
5227:
5219:
5212:
5209:
5202:
5196:
5192:
5182:
5173:
5169:
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5152:
5148:
5142:
5138:
5115:
5112:
5106:
5103:
5097:
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5079:
5076:
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5060:
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5054:
5051:
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5039:
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5029:
5023:
5020:
5017:
5014:
5011:
5008:
5005:
4982:
4979:
4963:
4947:
4936:
4932:
4927:
4921:
4918:
4860:
4857:
4854:
4851:
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4845:
4842:
4839:
4836:
4833:
4830:
4826:
4820:
4816:
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4777:
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4609:
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4577:
4567:
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4542:
4533:
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4526:
4523:
4503:
4500:
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4491:
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4485:
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4470:
4465:
4462:
4456:
4453:
4450:
4442:
4438:
4434:
4429:
4424:
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4413:
4410:
4407:
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4398:
4395:
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4075:
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4069:
4066:
4063:
4060:
4055:
4052:
4047:
4041:
4038:
4033:
4030:
4024:
4014:voltage source
4012:attached to a
4005:
4002:
4000:
3997:
3996:
3995:
3991:itself is the
3971:
3968:
3964:
3959:
3956:
3953:
3943:
3942:
3920:
3917:
3911:
3902:
3891:
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3823:
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3795:
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3764:
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3755:
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3735:
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3709:
3705:
3698:
3680:
3669:
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3656:
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3590:
3580:
3568:
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3546:
3523:
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2082:
2078:
2074:
2069:
2064:
2060:
2053:
2041:
2038:
2026:
2021:
2018:
2013:
2004:
1997:
1992:
1989:
1984:
1975:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1907:
1904:
1899:
1896:
1890:
1878:
1875:
1873:
1870:
1857:
1837:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1764:
1760:
1757:
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1716:
1713:
1710:
1695:
1692:
1680:
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1668:
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1602:
1598:
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1570:
1565:
1560:
1554:
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1537:
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1525:
1521:
1515:
1511:
1505:
1499:
1496:
1492:
1479:
1475:
1470:
1464:
1461:
1457:
1449:
1446:
1441:
1438:
1432:
1426:
1423:
1419:
1395:
1384:
1380:
1375:
1369:
1366:
1361:
1358:
1352:
1349:
1346:
1337:
1333:
1330:
1327:
1324:
1315:
1311:
1308:
1305:
1293:
1290:
1267:
1258:
1254:
1251:
1248:
1245:
1234:
1230:
1225:
1222:
1197:
1193:
1190:
1187:
1184:
1173:
1169:
1164:
1161:
1141:
1134:
1099:
1096:
1095:
1094:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1056:
1053:
1048:
1045:
1028:
1027:
1018:
1016:
1005:
1000:
997:
992:
983:
979:
976:
973:
961:
957:
953:
948:
936:
932:
921:
917:
901:
900:
899:
898:
890:
889:
880:
878:
867:
862:
859:
854:
845:
841:
838:
835:
830:
820:
816:
800:
799:
795:
780:
769:
758:
755:
752:
749:
746:
734:
730:
719:
715:
709:
706:
701:
691:
687:
681:
675:
672:
667:
664:
650:
639:
636:
633:
630:
627:
624:
613:
610:
607:
601:
591:
582:
578:
575:
572:
569:
566:
563:
554:
550:
547:
541:
538:
533:
530:
516:
506:
500:
474:
470:
467:
464:
458:
449:
445:
442:
439:
428:
405:
402:
399:
396:
393:
390:
387:
384:
381:
378:
372:
369:
364:
361:
355:
336:
326:
325:
313:
299:
287:
249:
246:
242:
241:
238:
235:
232:
229:
221:
218:
178:
175:
156:
155:
150:
145:
45:by a suitable
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6377:
6366:
6363:
6362:
6360:
6350:
6349:
6344:
6341:
6338:
6337:
6332:
6329:
6326:
6323:
6322:
6318:
6303:
6299:
6293:
6290:
6283:
6278:
6275:
6273:
6270:
6268:
6265:
6263:
6260:
6258:
6255:
6253:
6250:
6248:
6247:Natural units
6245:
6243:
6240:
6238:
6235:
6234:
6229:
6227:
6225:
6221:
6217:
6213:
6209:
6207:
6206:normalization
6202:
6198:
6192:
6184:
6182:
6169:
6157:
6142:
6136:
6133:
6121:
6106:
6099:
6093:
6083:
6076:
6068:
6058:
6051:
6045:
6041:
6035:
6031:
6025:
6021:
6007:
6003:
5981:
5958:
5949:
5941:
5937:
5928:
5925:
5905:
5893:
5878:
5866:
5860:
5848:
5833:
5826:
5820:
5810:
5803:
5795:
5785:
5778:
5772:
5768:
5762:
5758:
5748:
5735:
5728:
5725:
5715:
5706:
5699:
5696:
5689:
5678:
5669:
5663:
5659:
5653:
5649:
5623:
5613:
5600:
5587:
5575:
5560:
5550:
5541:
5536:
5527:
5523:
5520:
5514:
5502:
5487:
5480:
5474:
5464:
5453:
5442:
5433:
5427:
5423:
5417:
5413:
5406:
5398:
5388:
5381:
5375:
5371:
5365:
5361:
5345:
5330:
5323:
5320:
5308:
5293:
5286:
5280:
5270:
5261:
5252:
5246:
5242:
5238:
5233:
5230:
5225:
5217:
5207:
5200:
5194:
5190:
5180:
5171:
5167:
5159:
5156:
5150:
5140:
5136:
5126:
5113:
5101:
5086:
5080:
5065:
5058:
5052:
5049:
5037:
5027:
5018:
5015:
5009:
5003:
4977:
4962:
4945:
4934:
4930:
4925:
4916:
4903:
4899:
4892:
4885:
4881:
4876:
4871:
4858:
4852:
4846:
4843:
4840:
4834:
4828:
4824:
4818:
4814:
4808:
4804:
4800:
4795:
4792:
4787:
4779:
4775:
4771:
4765:
4761:
4752:
4749:
4743:
4733:
4729:
4720:
4716:
4708:
4703:
4701:
4697:
4684:
4681:
4672:
4668:
4659:
4653:
4650:
4644:
4641:
4638:
4635:
4629:
4624:
4621:
4616:
4607:
4600:
4595:
4591:
4587:
4584:
4575:
4565:
4556:
4552:
4549:
4546:
4540:
4531:
4527:
4524:
4521:
4498:
4489:
4486:
4483:
4480:
4477:
4471:
4468:
4463:
4460:
4454:
4451:
4448:
4440:
4436:
4432:
4427:
4422:
4418:
4405:
4402:
4396:
4393:
4388:
4384:
4380:
4375:
4372:
4367:
4361:
4358:
4353:
4350:
4344:
4341:
4333:
4329:
4325:
4320:
4315:
4311:
4304:
4296:
4292:
4288:
4284:
4276:
4274:
4272:
4271:time constant
4268:
4263:
4250:
4247:
4244:
4241:
4235:
4232:
4229:
4226:
4217:
4210:
4205:
4201:
4197:
4194:
4185:
4178:
4169:
4165:
4162:
4159:
4153:
4144:
4140:
4137:
4134:
4111:
4105:
4102:
4099:
4096:
4090:
4087:
4082:
4079:
4067:
4061:
4058:
4053:
4050:
4045:
4039:
4036:
4031:
4028:
4022:
4015:
4011:
4008:For a series
4003:
3998:
3994:
3993:damping ratio
3990:
3987:
3986:
3985:
3969:
3966:
3962:
3957:
3954:
3951:
3940:
3936:
3935:
3934:
3918:
3915:
3909:
3900:
3885:
3881:
3880:
3879:
3866:
3861:
3856:
3852:
3846:
3837:
3822:
3818:
3817:
3816:
3814:
3809:
3793:
3790:
3784:
3781:
3776:
3772:
3768:
3765:
3762:
3759:
3753:
3750:
3745:
3742:
3736:
3733:
3725:
3721:
3717:
3712:
3707:
3703:
3696:
3686:
3679:
3675:
3654:
3647:
3632:
3625:
3610:
3603:
3588:
3581:
3566:
3559:
3544:
3537:
3536:
3535:
3528:
3521:
3519:
3518:derivatives.
3512:
3510:
3508:
3502:
3500:
3499:superposition
3496:
3492:
3488:
3484:
3479:
3477:
3473:
3469:
3466:The function
3464:
3451:
3445:
3439:
3436:
3433:
3427:
3421:
3416:
3401:
3398:
3394:
3380:
3376:
3370:
3365:
3362:
3359:
3355:
3351:
3345:
3339:
3334:
3330:
3326:
3320:
3314:
3308:
3305:
3301:
3294:
3290:
3286:
3283:
3280:
3274:
3268:
3260:
3257:
3254:
3250:
3246:
3240:
3237:
3234:
3230:
3222:
3219:
3216:
3212:
3208:
3202:
3196:
3188:
3184:
3180:
3174:
3170:
3162:
3158:
3149:
3141:
3139:
3126:
3120:
3114:
3111:
3108:
3105:
3099:
3096:
3091:
3088:
3082:
3079:
3076:
3068:
3064:
3060:
3055:
3050:
3046:
3034:
3030:
3026:
3022:
3018:
3017:damping ratio
3014:
3010:
3005:
2992:
2986:
2983:
2979:
2954:
2944:
2941:
2932:
2919:
2913:
2907:
2904:
2901:
2898:
2892:
2889:
2884:
2881:
2872:
2869:
2865:
2860:
2852:
2848:
2844:
2839:
2834:
2830:
2817:
2804:
2799:
2796:
2791:
2782:
2766:
2762:
2758:
2753:
2741:
2737:
2730:
2721:
2717:
2711:
2708:
2697:
2674:
2671:
2665:
2656:
2647:
2630:
2627:
2622:
2613:
2604:
2603:
2602:
2597:
2589:
2587:
2582:
2575:
2570:
2557:
2551:
2545:
2533:
2529:
2522:
2511:
2505:
2499:
2496:
2491:
2488:
2481:
2470:
2464:
2458:
2455:
2450:
2447:
2439:
2436:
2425:
2421:
2413:
2409:
2405:
2400:
2395:
2391:
2379:
2376:
2363:
2357:
2351:
2348:
2345:
2333:
2329:
2323:
2320:
2317:
2314:
2305:
2301:
2298:
2292:
2289:
2284:
2281:
2267:
2257:
2251:
2248:
2240:
2236:
2232:
2227:
2222:
2218:
2207:
2196:
2184:
2178:
2170:
2168:
2164:
2156:
2154:
2141:
2135:
2129:
2126:
2123:
2120:
2117:
2114:
2108:
2105:
2100:
2097:
2091:
2088:
2080:
2076:
2072:
2067:
2062:
2058:
2051:
2039:
2037:
2024:
2019:
2016:
2011:
2002:
1995:
1990:
1987:
1982:
1973:
1964:
1963:
1958:
1957:
1951:
1938:
1932:
1926:
1923:
1920:
1917:
1914:
1911:
1905:
1902:
1897:
1894:
1888:
1876:
1871:
1869:
1855:
1835:
1826:
1813:
1807:
1801:
1798:
1789:
1783:
1777:
1774:
1762:
1758:
1752:
1749:
1743:
1737:
1714:
1708:
1701:
1693:
1691:
1678:
1670:
1666:
1662:
1656:
1652:
1642:
1631:
1625:
1620:
1615:
1610:
1603:
1600:
1596:
1583:
1579:
1573:
1568:
1563:
1558:
1552:
1549:
1545:
1540:
1535:
1527:
1523:
1519:
1513:
1509:
1497:
1494:
1490:
1477:
1473:
1468:
1462:
1459:
1455:
1447:
1444:
1439:
1436:
1430:
1424:
1421:
1417:
1406:
1393:
1382:
1378:
1373:
1367:
1364:
1359:
1356:
1347:
1344:
1335:
1331:
1328:
1325:
1313:
1309:
1306:
1303:
1291:
1289:
1287:
1283:
1278:
1265:
1256:
1252:
1249:
1246:
1232:
1228:
1223:
1220:
1195:
1191:
1188:
1185:
1171:
1167:
1162:
1159:
1151:
1147:
1140:
1133:
1129:
1125:
1121:
1117:
1113:
1109:
1105:
1098:Substitutions
1097:
1081:
1075:
1069:
1066:
1063:
1060:
1054:
1051:
1046:
1043:
1032:
1031:
1026:
1019:
1017:
1003:
998:
995:
990:
981:
974:
971:
959:
955:
951:
946:
934:
930:
919:
915:
905:
904:
897:Subsequently,
896:
895:
894:
893:
888:
881:
879:
865:
860:
857:
852:
843:
836:
833:
828:
818:
814:
804:
803:
794:
778:
770:
756:
750:
744:
732:
728:
717:
713:
707:
704:
699:
689:
685:
679:
673:
670:
665:
662:
651:
637:
631:
625:
622:
599:
580:
576:
570:
567:
564:
561:
552:
548:
545:
539:
536:
531:
528:
514:
504:
498:
472:
468:
465:
462:
456:
447:
443:
440:
437:
429:
426:
422:
418:
417:
416:
403:
397:
391:
388:
385:
382:
379:
376:
370:
367:
362:
359:
353:
345:
340:
335:
331:
311:
303:
300:
285:
277:
274:
273:
272:
269:
267:
263:
259:
255:
247:
245:
239:
236:
233:
230:
227:
226:
225:
219:
217:
215:
211:
210:natural units
207:
203:
197:
195:
191:
188:
184:
176:
174:
171:
167:
165:
161:
160:normalization
154:
151:
149:
146:
144:
141:
140:
139:
135:
133:
129:
128:mean lifetime
125:
121:
117:
113:
112:doubling time
109:
105:
101:
100:time constant
97:
93:
88:
86:
82:
78:
74:
70:
69:
64:
60:
56:
52:
48:
44:
40:
36:
32:
19:
6345:
6333:
6305:. Retrieved
6301:
6292:
6204:
6194:
6002:ground state
5749:
5601:
5127:
4996:defined via
4960:
4901:
4897:
4890:
4883:
4879:
4875:wavefunction
4872:
4712:
4698:
4294:
4290:
4286:
4282:
4280:
4264:
4007:
3988:
3944:
3938:
3892:
3883:
3829:
3820:
3810:
3684:
3677:
3673:
3670:
3533:
3516:
3503:
3480:
3471:
3467:
3465:
3147:
3146:The general
3145:
3024:
3012:
3008:
3006:
2933:
2818:
2695:
2693:
2595:
2593:
2580:
2573:
2571:
2380:
2377:
2171:
2166:
2162:
2160:
2043:
1960:
1954:
1952:
1880:
1827:
1697:
1407:
1295:
1285:
1281:
1279:
1149:
1145:
1138:
1131:
1127:
1123:
1119:
1111:
1107:
1103:
1101:
1020:
882:
792:
424:
420:
341:
333:
329:
327:
301:
275:
270:
265:
261:
257:
253:
251:
243:
223:
213:
205:
201:
198:
193:
189:
180:
168:
157:
136:
131:
89:
72:
67:
66:
51:parameterize
30:
29:
6257:RLC circuit
3489:are either
248:Conventions
65:, the term
6307:2024-08-23
6284:References
6267:RC circuit
6262:RL circuit
6197:statistics
181:Suppose a
164:statistics
41:involving
6161:~
6146:~
6143:ψ
6125:~
6110:~
6107:ψ
6087:~
6062:~
6036:−
6022:ω
6019:ℏ
5985:~
5953:~
5938:ω
5935:ℏ
5929:≡
5897:~
5882:~
5879:ψ
5870:~
5852:~
5837:~
5834:ψ
5814:~
5789:~
5763:−
5729:ω
5722:ℏ
5703:⇒
5686:ℏ
5660:ω
5617:~
5579:~
5564:~
5561:ψ
5547:ℏ
5506:~
5491:~
5488:ψ
5468:~
5450:ℏ
5424:ω
5392:~
5366:−
5358:⇒
5349:~
5334:~
5331:ψ
5312:~
5297:~
5294:ψ
5274:~
5243:ω
5211:~
5147:ℏ
5141:−
5105:~
5090:~
5087:ψ
5053:ψ
5031:~
5019:ψ
5004:ψ
4981:~
4978:ψ
4926:≡
4920:~
4847:ψ
4829:ψ
4805:ω
4740:ℏ
4734:−
4682:ω
4666:Ω
4639:ζ
4553:τ
4528:χ
4499:τ
4496:Ω
4490:
4481:χ
4472:τ
4464:χ
4455:ζ
4437:τ
4428:χ
4412:⇒
4403:ω
4397:
4166:τ
4141:χ
4112:τ
4100:χ
4091:τ
4083:χ
4074:⇒
3955:ζ
3791:ω
3785:
3356:∑
3284:…
3258:−
3238:−
3220:−
3121:τ
3109:χ
3100:τ
3092:χ
3083:ζ
3065:τ
3056:χ
3029:linewidth
2945:ζ
2914:τ
2902:χ
2893:τ
2885:χ
2849:τ
2840:χ
2779:⇒
2552:τ
2497:χ
2459:τ
2451:χ
2410:τ
2401:χ
2358:τ
2330:τ
2315:χ
2293:τ
2285:χ
2237:τ
2228:χ
1856:τ
1808:τ
1790:τ
1759:τ
1667:τ
1604:τ
1504:⇒
1498:τ
1463:τ
1440:τ
1360:τ
1351:⇒
1348:τ
1323:⇒
1310:τ
1253:χ
1244:⇒
1221:χ
1192:τ
1183:⇒
1160:τ
1076:τ
1064:χ
1055:τ
1047:χ
978:⇒
840:⇒
779:χ
751:τ
705:χ
674:τ
666:χ
632:τ
577:τ
562:χ
540:τ
532:χ
469:τ
444:χ
312:χ
286:τ
177:Rationale
120:half-life
77:intrinsic
6359:Category
6230:See also
1116:function
1110:, where
183:pendulum
55:measured
39:equation
37:from an
6214:by the
3579:= time
3485:of its
1122:. Both
256:" and "
68:scaling
63:systems
4958:where
4663:
4633:
4604:
4572:
4569:
4544:
4267:charge
4239:
4214:
4182:
4157:
2975:
2948:
1999:
620:
593:
460:
214:nature
206:system
187:period
96:length
3493:, or
3483:roots
1962:Eq. 2
1956:Eq. 1
1730:then
1114:is a
208:with
98:, or
6222:and
4713:The
3683:cos(
3491:real
2579:and
2165:and
1959:and
1284:and
1148:and
1137:and
1126:and
430:Set
6195:In
4721:is
4487:cos
4394:cos
3782:cos
3023:. 2
2648:If
2605:If
1118:of
212:of
162:in
6361::
6300:.
6226:.
4904:)|
4886:)|
4289:,
4285:,
4010:RC
3685:ωt
3676:=
3509:.
3501:.
3478:.
3035:.
2601::
1868:.
346::
166:.
94:,
81:SI
6351:.
6310:.
6208:.
6170:.
6167:)
6158:x
6152:(
6137:E
6134:=
6131:)
6122:x
6116:(
6100:)
6094:2
6084:x
6077:+
6069:2
6059:x
6052:d
6046:2
6042:d
6032:(
6026:2
5982:E
5959:.
5950:E
5942:2
5926:E
5906:,
5903:)
5894:x
5888:(
5867:E
5861:=
5858:)
5849:x
5843:(
5827:)
5821:2
5811:x
5804:+
5796:2
5786:x
5779:d
5773:2
5769:d
5759:(
5736:.
5726:m
5716:=
5711:c
5707:x
5700:1
5697:=
5690:2
5679:4
5674:c
5670:x
5664:2
5654:2
5650:m
5624:2
5614:x
5588:.
5585:)
5576:x
5570:(
5551:2
5542:E
5537:2
5532:c
5528:x
5524:m
5521:2
5515:=
5512:)
5503:x
5497:(
5481:)
5475:2
5465:x
5454:2
5443:4
5438:c
5434:x
5428:2
5418:2
5414:m
5407:+
5399:2
5389:x
5382:d
5376:2
5372:d
5362:(
5355:)
5346:x
5340:(
5324:E
5321:=
5318:)
5309:x
5303:(
5287:)
5281:2
5271:x
5262:2
5257:c
5253:x
5247:2
5239:m
5234:2
5231:1
5226:+
5218:2
5208:x
5201:d
5195:2
5191:d
5181:2
5176:c
5172:x
5168:1
5160:m
5157:2
5151:2
5137:(
5114:.
5111:)
5102:x
5096:(
5081:=
5078:)
5075:)
5070:c
5066:x
5062:(
5059:x
5056:(
5050:=
5047:)
5042:c
5038:x
5028:x
5022:(
5016:=
5013:)
5010:x
5007:(
4964:c
4961:x
4946:,
4939:c
4935:x
4931:x
4917:x
4902:x
4900:(
4898:ψ
4896:|
4891:x
4884:x
4882:(
4880:ψ
4878:|
4859:.
4856:)
4853:x
4850:(
4844:E
4841:=
4838:)
4835:x
4832:(
4825:)
4819:2
4815:x
4809:2
4801:m
4796:2
4793:1
4788:+
4780:2
4776:x
4772:d
4766:2
4762:d
4753:m
4750:2
4744:2
4730:(
4685:.
4677:c
4673:t
4669:=
4660:,
4654:L
4651:C
4645:R
4642:=
4636:2
4630:,
4625:C
4622:L
4617:=
4612:c
4608:t
4601:,
4596:0
4592:V
4588:C
4585:=
4580:c
4576:x
4566:,
4561:c
4557:t
4550:=
4547:t
4541:,
4536:c
4532:x
4525:=
4522:Q
4502:)
4493:(
4484:=
4478:+
4469:d
4461:d
4452:2
4449:+
4441:2
4433:d
4423:2
4419:d
4409:)
4406:t
4400:(
4389:0
4385:V
4381:=
4376:C
4373:Q
4368:+
4362:t
4359:d
4354:Q
4351:d
4345:R
4342:+
4334:2
4330:t
4326:d
4321:Q
4316:2
4312:d
4305:L
4295:Q
4291:L
4287:C
4283:R
4251:.
4248:V
4245:=
4242:F
4236:,
4233:C
4230:R
4227:=
4222:c
4218:t
4211:,
4206:0
4202:V
4198:C
4195:=
4190:c
4186:x
4179:,
4174:c
4170:t
4163:=
4160:t
4154:,
4149:c
4145:x
4138:=
4135:Q
4115:)
4109:(
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1004:.
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638:.
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626:F
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612:f
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546:+
537:d
529:d
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441:=
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427:.
425:x
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404:.
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386:=
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330:x
324:.
302:x
298:.
276:t
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262:x
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254:x
194:T
190:T
132:e
20:)
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