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{{Hatnote|For the film of the same name, see [[Uncertainty (film)]].}}
{{more footnotes|date=June 2013}}
{{Certainty}}
[[Image:Double-Pendulum.svg|upright|thumb|A double pendulum consists of two [[pendulum]]s attached end to end.]]
[[File:Blank Fork.png|thumb|We are frequently presented with situations wherein a decision must be made when we are uncertain of exactly how to proceed.]]
In [[physics]] and [[mathematics]], in the area of [[dynamical systems]], a '''double pendulum''' is a [[pendulum]] with another pendulum attached to its end, and is a simple [[physical system]] that exhibits rich [[dynamical systems|dynamic behavior]] with a strong sensitivity to initial conditions.<ref>Levien RB and Tan SM. Double Pendulum: An experiment in chaos.''American Journal of Physics'' 1993; 61 (11): 1038</ref> The motion of a double pendulum is governed by a set of coupled [[ordinary differential equation]]s. For certain [[energy|energies]] its motion is [[chaos theory|chaotic]].
'''Uncertainty''' is a term used in subtly different ways in a number of fields, including [[philosophy]], [[physics]], [[statistics]], [[economics]], [[finance]], [[insurance]], [[psychology]], [[sociology]], [[engineering]], and [[information science]]. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in [[partially observable]] and/or [[stochastic]] environments, as well as due to [[ignorance]] and/or [[Laziness|indolence]].<ref>[[Peter Norvig]], [[Sebastian Thrun]]. [[Udacity]]: [https://www.udacity.com/wiki/cs271/unit1_notes Introduction to Artificial Intelligence]</ref>


==Concepts==
==Analysis and interpretation==
Although the terms are used in various ways among the general public, many specialists in [[decision theory]], [[statistics]] and other quantitative fields have defined uncertainty, risk, and their measurement as:
Several variants of the double pendulum may be considered; the two limbs may be of equal or unequal lengths and masses, they may be [[simple pendulum]]s or [[compound pendulum]]s (also called complex pendulums) and the motion may be in three dimensions or restricted to the vertical plane. In the following analysis, the limbs are taken to be identical compound pendulums of length <math>\ell</math> and mass <math>m</math>, and the motion is restricted to two dimensions.
# '''Uncertainty''': The lack of certainty. A state of having limited knowledge where it is impossible to exactly describe the existing state, a future outcome, or more than one possible outcome.
# '''Measurement of Uncertainty''': A set of possible states or outcomes where probabilities are assigned to each possible state or outcome – this also includes the application of a probability density function to continuous variable
# '''Risk''': A state of uncertainty where some possible outcomes have an undesired effect or significant loss.
# '''Measurement of Risk''': A set of measured uncertainties where some possible outcomes are losses, and the magnitudes of those losses – this also includes loss functions over continuous variables.<ref>• Douglas Hubbard (2010). ''How to Measure Anything: Finding the Value of Intangibles in Business'', 2nd ed. John Wiley & Sons. [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470539399.html Description], [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470539399,descCd-tableOfContents.html contents], and  [http://books.google.com/books?hl=en&lr=&id=UFAkkGaY1x4C&oi=fnd&pg=PR5&ots=Jm_WeKJYwO&sig=JeG1WGp5GhVHyHiUD9DWmWzLfwg#v=onepage&q&f=false preview].<br />&nbsp;&nbsp; • [[Jean-Jacques Laffont]] (1989). ''The Economics of Uncertainty and Information'', MIT Press. [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=7640 Description] and chapter-preview [http://books.google.com/books/p/harvard?id=7r484x3HVu4C&printsec=find&pg=PR5=#v=onepage&q&f=false links].<br />&nbsp;&nbsp; • _____ (1980). ''Essays in the Economics of Uncertainty'', Harvard University Press. Chapter-preview [http://books.google.com/books/p/harvard?id=8wwbolpmLH8C&printsec=find&pg=PR7#v=onepage&q&f=false links].<br />&nbsp;&nbsp; • Robert G. Chambers and [[John Quiggin]] (2000). ''Uncertainty, Production, Choice, and Agency: The State-Contingent Approach''. Cambridge. [http://www.cambridge.org/aus/catalogue/catalogue.asp?isbn=9780521622448 Description] and  [http://books.google.com/books?id=_R54pqQWvPYC&pg=PR7lpg=PR7&dq=&source=bl&ots=6oKu2mnosK&sig=br2OLdOohXfbBB9UT5icGmj0imo&hl=en&ei=5Ow6TYq1DML6lwe61eCCBw&sa=X&oi=book_result&ct=result&resnum=4&ved=0CCwQ6AEwAw#v=onepage&q&f=false preview.] ISBN 0-521-62244-1</ref>


'''[[Knightian uncertainty]]'''. In his seminal work ''Risk, Uncertainty, and Profit'' (1921), [[University of Chicago]] economist [[Frank Knight]] established the important distinction between [[risk]] and uncertainty'':''<ref>{{cite book |last=Knight |first=F. H. |year=1921 |title=Risk, Uncertainty, and Profit |location=Boston |publisher=Hart, Schaffner & Marx }}</ref>
[[Image:Double-compound-pendulum-dimensioned.svg|right|thumb|Double compound pendulum]]
{{Cquote|Uncertainty must be taken in a sense radically distinct from the familiar notion of risk, from which it has never been properly separated.... The essential fact is that 'risk' means in some cases a quantity susceptible of measurement, while at other times it is something distinctly not of this character; and there are far-reaching and crucial differences in the bearings of the phenomena depending on which of the two is really present and operating.... It will appear that a measurable uncertainty, or 'risk' proper, as we shall use the term, is so far different from an unmeasurable one that it is not in effect an uncertainty at all.}}{{Cquote|You cannot be certain about uncertainty.}}
In a compound pendulum, the mass is distributed along its length. If the mass is evenly distributed, then the center of mass of each limb is at its midpoint, and the limb has a [[moment of inertia]] of <math>\textstyle I=\frac{1}{12} m \ell^2</math> about that point.<!--  The moment of inertia of a rod rotating around an axis attached to one of its ends equals <math>\textstyle I=\frac{1}{3} m \ell^2</math>. -->


There are other taxonomies of uncertainties and decisions that include a broader sense of uncertainty and how it should be approached from an ethics perspective:<ref name="embo1">{{cite journal |author=Tannert C, Elvers HD, Jandrig B |title=The ethics of uncertainty. In the light of possible dangers, research becomes a moral duty. |journal=EMBO Rep. |volume=8 |issue=10 |pages=892–6 |year=2007 |pmid=17906667 |doi= 10.1038/sj.embor.7401072 |pmc=2002561}}</ref>
It is convenient to use the angles between each limb and the vertical as the [[generalized coordinates]] defining the [[configuration space|configuration]] of the system. These angles are denoted θ<sub>1</sub> and θ<sub>2</sub>. The position of the center of mass of each rod may be written in terms of these two coordinates. If the origin of the [[Cartesian coordinate system]] is taken to be at the point of suspension of the first pendulum, then the center of mass of this pendulum is at:
:<math>
x_1 = \frac{\ell}{2} \sin \theta_1,
</math>
:<math>
y_1 = -\frac{\ell}{2} \cos \theta_1
</math>
and the center of mass of the second pendulum is at
:<math>
x_2 = \ell \left (  \sin \theta_1 + \frac{1}{2} \sin \theta_2 \right ),
</math>
:<math>
y_2 = -\ell \left (  \cos \theta_1 + \frac{1}{2} \cos \theta_2 \right ).
</math>
This is enough information to write out the Lagrangian.


[[File:Uncertainty.svg|thumb|center|550px|A taxonomy of uncertainty]]
===Lagrangian===
The [[Lagrangian]] is
:<math>
\begin{align}L & = \mathrm{Kinetic~Energy} - \mathrm{Potential~Energy} \\
              & = \frac{1}{2} m \left ( v_1^2 + v_2^2 \right ) + \frac{1}{2} I \left ( {\dot \theta_1}^2 + {\dot \theta_2}^2 \right ) - m g \left ( y_1 + y_2 \right ) \\
              & = \frac{1}{2} m \left ( {\dot x_1}^2 + {\dot y_1}^2 + {\dot x_2}^2 + {\dot y_2}^2 \right ) + \frac{1}{2} I \left ( {\dot \theta_1}^2 + {\dot \theta_2}^2 \right ) - m g \left ( y_1 + y_2 \right ) \end{align}
</math>
The first term is the ''linear'' [[kinetic energy]] of the [[center of mass]] of the bodies and the second term is the ''rotational'' kinetic energy around the center of mass of each rod. The last term is the [[potential energy]] of the bodies in a uniform gravitational field. The [[Newton's notation|dot-notation]] indicates the [[time derivative]] of the variable in question.


{{Quote box |quoted=true |bgcolor=#FFFFF0 |salign=center |quote=There are some things that you know to be true, and others that you know to be false; yet, despite this extensive knowledge that you have, there remain many things whose truth or falsity is not known to you. We say that you are uncertain about them. You are uncertain, to varying degrees, about everything in the future; much of the past is hidden from you; and there is a lot of the present about which you do not have full information. Uncertainty is everywhere and you cannot escape from it. |source=[[Dennis Lindley]], ''Understanding Uncertainty'' (2006) |width=33% |align=right}}
Substituting the coordinates above and rearranging the equation gives
:<math>
L = \frac{1}{6} m \ell^2 \left [ {\dot \theta_2}^2 + 4 {\dot \theta_1}^2 + 3 {\dot \theta_1} {\dot \theta_2} \cos (\theta_1-\theta_2) \right ] + \frac{1}{2} m g \ell \left ( 3 \cos \theta_1 + \cos \theta_2 \right ).
</math>


For example, if you do not know whether it will rain tomorrow, then you have a state of uncertainty. If you apply probabilities to the possible outcomes using weather forecasts or even just a [[calibrated probability assessment]], you have quantified the uncertainty. Suppose you quantify your uncertainty as a 90% chance of sunshine. If you are planning a major, costly, outdoor event for tomorrow then you have risk since there is a 10% chance of rain and rain would be undesirable. Furthermore, if this is a business event and you would lose $100,000 if it rains, then you have quantified the risk (a 10% chance of losing $100,000). These situations can be made even more realistic by quantifying light rain vs. heavy rain, the cost of delays vs. outright cancellation, etc.
[[Image:double-compound-pendulum.gif|right|frame|Motion of the double compound pendulum (from numerical integration of the equations of motion)]]
[[Image:DPLE.jpg|right|thumb|Long exposure of double pendulum exhibiting chaotic motion (tracked with an [[LED]])]]
There is only one conserved quantity (the energy), and no conserved [[generalized momentum|momenta]]. The two momenta may be written as


Some may represent the risk in this example as the "expected opportunity loss" (EOL) or the chance of the loss multiplied by the amount of the loss (10% × $100,000 = $10,000). That is useful if the organizer of the event is "risk neutral", which most people are not. Most would be willing to pay a premium to avoid the loss. An [[insurance]] company, for example, would compute an EOL as a minimum for any insurance coverage, then add onto that other operating costs and profit. Since many people are willing to buy insurance for many reasons, then clearly the EOL alone is not the perceived value of avoiding the risk.
:<math>
p_{\theta_1} = \frac{\partial L}{\partial {\dot \theta_1}} = \frac{1}{6} m \ell^2 \left [ 8 {\dot \theta_1}  + 3 {\dot \theta_2} \cos (\theta_1-\theta_2) \right ]
</math>
and
:<math>
p_{\theta_2} = \frac{\partial L}{\partial {\dot \theta_2}} = \frac{1}{6} m \ell^2 \left [ 2 {\dot \theta_2} + 3 {\dot \theta_1} \cos (\theta_1-\theta_2) \right ].
</math>


Quantitative uses of the terms uncertainty and risk are fairly consistent from fields such as [[probability theory]], [[actuarial science]], and [[information theory]]. Some also create new terms without substantially changing the definitions of uncertainty or risk. For example, [[surprisal]] is a variation on uncertainty sometimes used in [[information theory]]. But outside of the more mathematical uses of the term, usage may vary widely. In [[cognitive psychology]], uncertainty can be real, or just a matter of perception, such as [[Expectation (epistemic)|expectations]], threats, etc.
These expressions may be inverted to get


Vagueness or ambiguity are sometimes described as "second order uncertainty", where there is uncertainty even about the definitions of uncertain states or outcomes. The difference here is that this uncertainty is about the human definitions and concepts, not an objective fact of nature. It has been argued that ambiguity, however, is always avoidable while uncertainty (of the "first order" kind) is not necessarily avoidable.
:<math>
{\dot \theta_1} = \frac{6}{m\ell^2} \frac{ 2 p_{\theta_1} - 3 \cos(\theta_1-\theta_2) p_{\theta_2}}{16 - 9 \cos^2(\theta_1-\theta_2)}
</math>
and
:<math>
{\dot \theta_2} = \frac{6}{m\ell^2} \frac{ 8 p_{\theta_2} - 3 \cos(\theta_1-\theta_2) p_{\theta_1}}{16 - 9 \cos^2(\theta_1-\theta_2)}.
</math>


Uncertainty may be purely a consequence of a lack of knowledge of obtainable facts. That is, you may be uncertain about whether a new rocket design will work, but this uncertainty can be removed with further analysis and experimentation. At the subatomic level, however, uncertainty may be a fundamental and unavoidable property of the universe. In [[quantum mechanics]], the [[Heisenberg Uncertainty Principle]] puts limits on how much an observer can ever know about the position and velocity of a particle. This may not just be ignorance of potentially obtainable facts but that there is no fact to be found. There is some controversy in physics as to whether such uncertainty is an irreducible property of nature or if there are "hidden variables" that would describe the state of a particle even more exactly than Heisenberg's uncertainty principle allows.
The remaining equations of motion are written as


==Measurements== <!-- Heavily linked section: Standard uncertainty, Concise notation -->
:<math>
{{Main|Measurement uncertainty}}
{\dot p_{\theta_1}} = \frac{\partial L}{\partial \theta_1} = -\frac{1}{2} m \ell^2 \left [ {\dot \theta_1} {\dot \theta_2} \sin (\theta_1-\theta_2) + 3 \frac{g}{\ell} \sin \theta_1 \right ]
{{See also|Uncertainty quantification|Uncertainty propagation}}
</math>
In [[metrology]], [[physics]], and [[engineering]], the uncertainty or [[margin of error]] of a measurement is stated by giving a range of values likely to enclose the true value. This may be denoted by [[error bar]]s on a graph, or by the following notations:
* ''measured value'' ± ''uncertainty
* ''measured value'' {{su|p=+uncertainty|b=−uncertainty}}
* ''measured value''(''uncertainty'')


The middle notation is used when the error is not symmetrical about the value – for example <math>3.4_{-0.2}^{+0.3}</math>. This can occur when using a logarithmic scale, for example. The latter "concise notation" is used for example by [[IUPAC]] in stating the [[list of elements by atomic mass|atomic mass]] of [[chemical element|elements]]. There, the uncertainty given in parenthesis applies to the [[significant figure|least significant figure]](s) of the number prior to the parenthesized value (i.e., counting from rightmost digit to left). For instance, {{val|1.00794|(7)}} stands for {{val|1.00794|0.00007}}, while {{val|1.00794|(72)}} stands for {{val|1.00794|0.00072}}.<ref>{{cite web|url=http://physics.nist.gov/cgi-bin/cuu/Info/Constants/definitions.html|title=Standard Uncertainty and Relative Standard Uncertainty|work=[[CODATA]] reference|publisher=[[NIST]]|accessdate=26 September 2011}}</ref>
and


Often, the uncertainty of a measurement is found by repeating the measurement enough times to get a good estimate of the [[standard deviation]] of the values. Then, any single value has an uncertainty equal to the standard deviation. However, if the values are averaged, then the mean measurement value has a much smaller uncertainty, equal to the [[standard error (statistics)|standard error]] of the mean, which is the standard deviation divided by the square root of the number of measurements. This procedure neglects [[systematic error]]s, however.
:<math>
{\dot p_{\theta_2}} = \frac{\partial L}{\partial \theta_2}
= -\frac{1}{2} m \ell^2 \left [ -{\dot \theta_1} {\dot \theta_2} \sin (\theta_1-\theta_2) +  \frac{g}{\ell} \sin \theta_2 \right ].
</math>


When the uncertainty represents the standard error of the measurement, then about 68.3% of the time, the true value of the measured quantity falls within the stated uncertainty range. For example, it is likely that for 31.7% of the atomic mass values given on the [[list of elements by atomic mass]], the true value lies outside of the stated range. If the width of the interval is doubled, then probably only 4.6% of the true values lie outside the doubled interval, and if the width is tripled, probably only 0.3% lie outside. These values follow from the properties of the [[normal distribution]], and they apply only if the measurement process produces normally distributed errors. In that case, the quoted [[standard error (statistics)|standard errors]] are easily converted to 68.3% ("one [[sigma]]"), 95.4% ("two sigma"), or 99.7% ("three sigma") [[confidence interval]]s.{{citation needed|date=September 2014}}
These last four equations are explicit formulae for the time evolution of the system given its current state. It is not possible to go further and integrate these equations analytically{{Citation needed|date=June 2011}}, to get formulae for θ<sub>1</sub> and θ<sub>2</sub> as functions of time. It is however possible to perform this integration numerically using the [[Runge–Kutta methods|Runge Kutta]] method or similar techniques.


In this context, uncertainty depends on both the [[accuracy and precision]] of the measurement instrument. The lower the accuracy and precision of an instrument, the larger the measurement uncertainty is. Notice that precision is often determined as the [[standard deviation]] of the repeated measures of a given value, namely using the same method described above to assess measurement uncertainty. However, this method is correct only when the instrument is accurate. When it is inaccurate, the uncertainty is larger than the [[standard deviation]] of the repeated measures, and it appears evident that the uncertainty does not depend only on instrumental precision.
==Chaotic motion==
[[Image:Double_pendulum_flips_graph.png|thumb|Graph of the time for the pendulum to flip over as a function of initial conditions]]


==Uncertainty and the media==
The double pendulum undergoes [[chaotic motion]], and shows a sensitive dependence on [[initial conditions]].  The image to the right shows the amount of elapsed time before the pendulum "flips over," as a function of initial conditions. Here, the initial value of θ<sub>1</sub> ranges along the ''x''-direction, from &minus;3 to 3. The initial value θ<sub>2</sub> ranges along the ''y''-direction, from &minus;3 to 3The colour of each pixel indicates whether either pendulum flips within <math>10\sqrt{\ell/g }</math> (green), within <math>100\sqrt{\ell/g  }</math> (red), <math>1000\sqrt{\ell/g }</math> (purple) or <math>10000\sqrt{\ell/g  }</math> (blue)Initial conditions that don't lead to a flip within <math>10000\sqrt{\ell/}</math> are plotted white.
Uncertainty in science, and science in general, is often interpreted much differently in the public sphere than in the scientific community.<ref name=zehr>Zehr, S. C. (1999)Scientists’ representation of uncertainty. In Friedman, S.M., Dunwoody, S., & Rogers, C. L. (Eds.), Communicating uncertainty: Media coverage of new and controversial science (3-21). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.</ref> This is due in part to the diversity of the public audience, and the tendency for scientists to misunderstand lay audiences and therefore not communicate ideas clearly and effectively.<ref name=zehr /> One example is explained by the [[information deficit model]]. Also, in the public realm, there are often many scientific voices giving input on a single topic.<ref name=zehr /> For example, depending on how an issue is reported in the public sphere, discrepancies between outcomes of multiple scientific studies due to methodological differences could be interpreted by the public as a lack of consensus in a situation where a consensus does in fact exist.<ref name=zehr /> This interpretation may have even been intentionally promoted, as scientific uncertainty may be managed to reach certain goalsFor example, global warming contrarian activists took the advice of [[Frank Luntz]] to frame [[global warming]] as an issue of scientific uncertainty, which was a precursor to the conflict frame used by journalists when reporting the issue.<ref>{{cite journal |last=Nisbet |first=M. |last2=Scheufele |first2=D. A. |year=2009 |title=What’s next for science communication? Promising directions and lingering distractions |journal=[[American Journal of Botany]] |volume=96 |issue=10 |pages=1767–1778 |doi=10.3732/ajb.0900041 }}</ref>


“Indeterminacy can be loosely said to apply to situations in which not all the parameters of the system and their interactions are fully known, whereas ignorance refers to situations in which it is not known what is not known”.<ref>{{cite journal |last=Shackley |first=S. |last2=Wynne |first2=B. |year=1996 |title=Representing uncertainty in global climate change science and policy: Boundary-ordering devices and authority |journal=Science, Technology, & Human Values |volume=21 |issue=3 |pages=275–302 |doi=10.1177/016224399602100302 }}</ref> These unknowns, indeterminacy and ignorance, that exist in science are often “transformed” into uncertainty when reported to the public in order to make issues more manageable, since scientific indeterminacy and ignorance are difficult concepts for scientists to convey without losing credibility.<ref name=zehr /> Conversely, uncertainty is often interpreted by the public as ignorance.<ref>{{cite journal |last=Somerville |first=R. C. |last2=Hassol |first2=S. J. |year=2011 |title=Communicating the science of climate change |journal=Physics Today |volume= |issue= |pages=48–53 }}</ref> The transformation of indeterminacy and ignorance into uncertainty may be related to the public’s misinterpretation of uncertainty as ignorance.
The boundary of the central white region is defined in part by energy conservation with the following curve:


Journalists often either inflate uncertainty (making the science seem more uncertain than it really is) or downplay uncertainty (making the science seem more certain than it really is).<ref name=stocking>{{cite book |last=Stocking |first=H. |year=1999 |chapter=How journalists deal with scientific uncertainty |editor1-last=Friedman |editor1-first=S. M. |editor2-last=Dunwoody |editor2-first=S. |editor3-last=Rogers |editor3-first=C. L. |title=Communicating Uncertainty: Media Coverage of New and Controversial Science |pages=23–41 |location=Mahwah, NJ |publisher=Lawrence Erlbaum |isbn=0-8058-2727-7 }}</ref> One way that journalists inflate uncertainty is by describing new research that contradicts past research without providing context for the change<ref name=stocking /> Other times, journalists give scientists with minority views equal weight as scientists with majority views, without adequately describing or explaining the state of scientific consensus on the issue.<ref name=stocking /> In the same vein, journalists often give non-scientists the same amount of attention and importance as scientists.<ref name=stocking />
:<math>
3 \cos \theta_1 + \cos \theta_2  = 2. \,
</math>


Journalists may downplay uncertainty by eliminating “scientists’ carefully chosen tentative wording, and by losing these caveats the information is skewed and presented as more certain and conclusive than it really is”.<ref name=stocking /> Also, stories with a single source or without any context of previous research mean that the subject at hand is presented as more definitive and certain than it is in reality.<ref name=stocking /> There is often a “product over process” approach to [[science journalism]] that aids, too, in the downplaying of uncertainty.<ref name=stocking /> Finally, and most notably for this investigation, when science is framed by journalists as a triumphant quest, uncertainty is erroneously framed as “reducible and resolvable”.<ref name=stocking />
Within the region defined by this curve, that is if


Some media routines and organizational factors affect the overstatement of uncertainty; other media routines and organizational factors help inflate the certainty of an issue. Because the general public (in the United States) generally trusts scientists, when science stories are covered without alarm-raising cues from special interest organizations (religious groups, environmental organization, political factions, etc.) they are often covered in a business related sense, in an economic-development frame or a social progress frame.<ref name=nisbet>{{cite journal |last=Nisbet |first=M. |last2=Scheufele |first2=D. A. |year=2007 |title=The Future of Public Engagement |journal=The Scientist |volume=21 |issue=10 |pages=38–44 |doi= }}</ref> The nature of these frames is to downplay or eliminate uncertainty, so when economic and scientific promise are focused on early in the issue cycle, as has happened with coverage of plant biotechnology and nanotechnology in the United States, the matter in question seems more definitive and certain.<ref name=nisbet />
:<math>
3 \cos \theta_1 + \cos \theta_2  > 2, \,
</math>


Sometimes, too, stockholders, owners, or advertising will pressure a media organization to promote the business aspects of a scientific issue, and therefore any uncertainty claims that may compromise the business interests are downplayed or eliminated.<ref name=stocking />
then it is energetically impossible for either pendulum to flip.  Outside this region, the pendulum can flip, but it is a complex question to determine when it will flip.  Similar behavior is observed for a double pendulum composed of two point masses rather than two rods with distributed mass.<ref>Alex Small, ''[https://12d82b32-a-62cb3a1a-s-sites.googlegroups.com/site/physicistatlarge/Computational%20Physics%20Sample%20Project-Alex%20Small-v1.pdf Sample Final Project: One Signature of Chaos in the Double Pendulum]'', (2013). A report produced as an example for students.  Includes a derivation of the equations of motion, and a comparison between the double pendulum with 2 point masses and the double pendulum with 2 rods.</ref>


==Applications==
The lack of a natural excitation frequency has led to the use of double pendulum systems in seismic resistance designs in buildings, where the building itself is the primary inverted pendulum, and a secondary mass is connected to complete the double pendulum.
{{unordered list
| Investing in [[financial market]]s such as the stock market.
| Uncertainty or [[error]] is used in science and engineering notation. Numerical values should only be expressed to those digits that are physically meaningful, which are referred to as [[significant figures]]. Uncertainty is involved in every measurement, such as measuring a distance, a temperature, etc., the degree depending upon the instrument or technique used to make the measurement. Similarly, uncertainty is propagated through calculations so that the calculated value has some degree of uncertainty depending upon the uncertainties of the measured values and the equation used in the calculation.<ref>{{cite journal |last=Gregory |first=Kent J. |last2=Bibbo |first2=Giovanni |last3=Pattison |first3=John E. |year=2005 |title=A Standard Approach to Measurement Uncertainties for Scientists and Engineers in Medicine |journal=Australasian Physical and Engineering Sciences in Medicine |volume=28 |issue=2 |pages=131–139 |doi=10.1007/BF03178705 }}</ref>
| Uncertainty is designed into [[game]]s, most notably in [[gambling]], where [[probability|chance]] is central to play.
| In [[scientific modelling]], in which the prediction of future events should be understood to have a range of expected values.
| In [[physics]], the Heisenberg [[uncertainty principle]] forms the basis of modern [[quantum mechanics]].
| In [[meteorology|weather forecasting]] it is now commonplace to include data on the degree of uncertainty in a [[weather forecast]].
| Uncertainty is often an important factor in [[economics]]. According to economist [[Frank Knight]], it is different from [[risk]], where there is a specific [[probability]] assigned to each outcome (as when flipping a fair coin). Uncertainty involves a situation that has unknown probabilities, while the estimated probabilities of possible outcomes need not add to unity.
| In [[entrepreneurship]]: New products, services, firms and even markets are often created in the absence of probability estimates. According to entrepreneurship research, expert entrepreneurs predominantly use experience based heuristics called [[effectuation]] (as opposed to [[causality]]) to overcome uncertainty.
| In [[metrology]], [[measurement uncertainty]] is a central concept quantifying the dispersion one may reasonably attribute to a measurement result. Such an uncertainty can also be referred to as a measurement [[error]]. In daily life, measurement uncertainty is often implicit ("He is 6 feet tall" give or take a few inches), while for any serious use an explicit statement of the measurement uncertainty is necessary. The expected measurement uncertainty of many [[measuring instruments]] (scales, oscilloscopes, force gages, rulers, thermometers, etc.) is often stated in the manufacturer's specification.
| [[Mobile phone radiation|Mobile phone radiation and health]]


The most commonly used procedure for calculating measurement uncertainty is described in the "Guide to the Expression of Uncertainty in Measurement" (GUM) published by [[ISO]]. A derived work is for example the [[National Institute for Standards and Technology]] (NIST) Technical Note 1297, "Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results", and the Eurachem/Citac publication "Quantifying Uncertainty in Analytical Measurement". The uncertainty of the result of a measurement generally consists of several components. The components are regarded as [[random variables]], and may be grouped into two categories according to the method used to estimate their numerical values:
==See also==
* Type A, those evaluated by [[statistical]] methods
* [[Double inverted pendulum]]
* Type B, those evaluated by other means, e.g., by assigning a [[probability distribution]]
* [[Pendulum (mathematics)]]
By propagating the [[variance]]s of the components through a function relating the components to the measurement result, the combined measurement uncertainty is given as the square root of the resulting variance. The simplest form is the [[standard deviation]] of a repeated observation.| Uncertainty has been a common theme in art, both as a thematic device (see, for example, the indecision of [[Hamlet]]), and as a quandary for the artist (such as [[Martin Creed]]'s difficulty with deciding what artworks to make).
* Mid-20th century physics textbooks use the term "Double Pendulum" to mean a single bob suspended from a string which is in turn suspended from a V-shaped string.  This type of [[pendulum]], which produces [[Lissajous curves]], is now referred to as a [[Blackburn pendulum]]. An artistic application of this can be seen here:  http://paulwainwrightphotography.com/pendulum_gallery.shtml .
 
| Uncertainty assessment is significantly important for managing oil reservoirs where decisions are made based on uncertain models/outcomes.  
Predictions of oil and gas production from subsurface reservoirs are always uncertain.<ref> History matching production data and uncertainty assessment with an efficient TSVD parameterization algorithm, Journal of Petroleum Science and Engineering, http://www.sciencedirect.com/science/article/pii/S0920410513003227</ref>
}}


==See also==
==Notes==
{{colbegin|2}}
{{reflist}}
* [[Applied Information Economics]]
* [[Buckley's chance]]
* [[Certainty]]
* [[Fuzzy set theory]]
* [[Dempster–Shafer theory]]
* [[Game theory]]
* [[Information entropy]]
* [[Interval finite element]]
* [[Morphological analysis (problem-solving)]]
* [[Propagation of uncertainty]]
* [[Randomness]]
* [[Schrödinger's cat]]
* [[Statistical mechanics]]
*[[Measurement uncertainty]]
* [[Uncertainty quantification]]
* [[Uncertainty tolerance]]
* [[Volatility, uncertainty, complexity and ambiguity]]
{{colend}}


==References==
==References==
{{reflist|30em}}
*{{cite book
 
| last = Meirovitch
==Further reading==
| first = Leonard
* {{cite book |title=Understanding Uncertainty |last=Lindley |first=Dennis V. |authorlink=Dennis Lindley |date=2006-09-11 |publisher=[[John Wiley & Sons|Wiley-Interscience]] |isbn=978-0-470-04383-7}}
| year = 1986
* {{cite book |title=Theory of Decision under Uncertainty |last=Gilboa |first=Itzhak |authorlink=Itzhak Gilboa |year=2009 |place=Cambridge |publisher=[[Cambridge University Press]] |isbn=9780521517324}}
| title = Elements of Vibration Analysis
* {{cite book |title=Reasoning about Uncertainty |last=Halpern |first=Joseph |authorlink=Joseph Halpern |date=2005-09-01 |publisher=[[MIT Press]] |isbn=9780521517324}}
| edition = 2nd edition
* {{cite book |title=Ignorance and Uncertainty |last=Smithson |first=Michael |authorlink=Michael Smithson |year=1989 |place=New York |publisher=[[Springer-Verlag]] |isbn=0-387-96945-4}}
| publisher = McGraw-Hill Science/Engineering/Math
| isbn = 0-07-041342-8
}}
* Eric W. Weisstein, ''[http://scienceworld.wolfram.com/physics/DoublePendulum.html Double pendulum]'' (2005), ScienceWorld ''(contains details of the complicated equations involved)'' and "[http://demonstrations.wolfram.com/DoublePendulum/ Double Pendulum]" by Rob Morris, [[Wolfram Demonstrations Project]], 2007 (animations of those equations).
* Peter Lynch, ''[http://www.maths.tcd.ie/~plynch/SwingingSpring/doublependulum.html Double Pendulum]'', (2001). ''(Java applet simulation.)''
* Northwestern University, ''[http://www.physics.northwestern.edu/vpl/mechanics/pendulum.html Double Pendulum]'', ''(Java applet simulation.)''
* Theoretical High-Energy Astrophysics Group at UBC, ''[http://tabitha.phas.ubc.ca/wiki/index.php/Double_pendulum Double pendulum]'', (2005).


==External links==
==External links==
{{Wiktionary|uncertainty}}
*Animations and explanations of a [http://www.physics.usyd.edu.au/~wheat/dpend_html/ double pendulum] and a [http://www.physics.usyd.edu.au/~wheat/sdpend/ physical double pendulum (two square plates)] by Mike Wheatland (Univ. Sydney)
{{wikiquote}}
*[http://www.youtube.com/watch?v=Uzlccwt5SKc&NR=1 Video] of a double square pendulum with three (almost) identical starting conditions.
* [http://www.springer.com/dal/home/generic/search/results?SGWID=1-40109-22-24419924-0 Measurement Uncertainties in Science and Technology, Springer 2005]
*Double pendulum physics simulation from [http://www.myphysicslab.com/dbl_pendulum.html www.myphysicslab.com]
* [http://www.uncertainty.de Proposal for a New Error Calculus]
*Simulation, equations and explanation of [http://www.chris-j.co.uk/rott.php Rott's pendulum]
* [http://www.uncertainty.de/p97_s.pdf  Estimation of Measurement Uncertainties — an Alternative to the ISO Guide]
*Comparison videos of a double pendulum with the same initial starting conditions on [http://www.youtube.com/watch?v=O2ySvbL3-yA YouTube]
* [http://www.fasor.com/iso25/bibliography_of_uncertainty.htm Bibliography of Papers Regarding Measurement Uncertainty]
* [http://freddie.witherden.org/tools/doublependulum/ Double Pendulum Simulator] - An open source simulator written in [[C++]] using the [[Qt (toolkit)|Qt toolkit]].
* [http://physics.nist.gov/Pubs/guidelines/contents.html Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results]
* [http://www.imaginary2008.de/cinderella/english/G2.html Online Java simulator] of the [[Imaginary_(exhibition)|Imaginary exhibition]].
* [http://strategic.mit.edu Strategic Engineering: Designing Systems and Products under Uncertainty (MIT Research Group)]
* Vadas Gintautas, [[Alfred Hubler|Alfred Hübler]] (2007). [http://pre.aps.org/abstract/PRE/v75/i5/e057201 Experimental evidence for mixed reality states in an interreality system] Phys. Rev. E 75, 057201 Presents data on an experimental, mixed reality system in which a real and virtual pendulum complexly interact.
* [http://understandinguncertainty.org/ Understanding Uncertainty site] from Cambridge's Winton programme
* {{cite web|last=Bowley|first=Roger|title=∆ – Uncertainty|url=http://www.sixtysymbols.com/videos/uncertainty.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|year=2009}}


{{Chaos theory}}


[[Category:Statistical theory]]
[[Category:Chaotic maps]]
[[Category:Statistical terminology]]
[[Category:Pendulums]]
[[Category:Experimental physics]]
[[Category:Measurement]]
[[Category:Cognition]]
[[Category:Probability interpretations]]
[[Category:Concepts in epistemology]]
[[Category:Prospect theory]]
[[Category:Doubt]]

Revision as of 20:30, 9 August 2014

Template:More footnotes

A double pendulum consists of two pendulums attached end to end.

In physics and mathematics, in the area of dynamical systems, a double pendulum is a pendulum with another pendulum attached to its end, and is a simple physical system that exhibits rich dynamic behavior with a strong sensitivity to initial conditions.[1] The motion of a double pendulum is governed by a set of coupled ordinary differential equations. For certain energies its motion is chaotic.

Analysis and interpretation

Several variants of the double pendulum may be considered; the two limbs may be of equal or unequal lengths and masses, they may be simple pendulums or compound pendulums (also called complex pendulums) and the motion may be in three dimensions or restricted to the vertical plane. In the following analysis, the limbs are taken to be identical compound pendulums of length and mass , and the motion is restricted to two dimensions.

Double compound pendulum

In a compound pendulum, the mass is distributed along its length. If the mass is evenly distributed, then the center of mass of each limb is at its midpoint, and the limb has a moment of inertia of about that point.

It is convenient to use the angles between each limb and the vertical as the generalized coordinates defining the configuration of the system. These angles are denoted θ1 and θ2. The position of the center of mass of each rod may be written in terms of these two coordinates. If the origin of the Cartesian coordinate system is taken to be at the point of suspension of the first pendulum, then the center of mass of this pendulum is at:

and the center of mass of the second pendulum is at

This is enough information to write out the Lagrangian.

Lagrangian

The Lagrangian is

The first term is the linear kinetic energy of the center of mass of the bodies and the second term is the rotational kinetic energy around the center of mass of each rod. The last term is the potential energy of the bodies in a uniform gravitational field. The dot-notation indicates the time derivative of the variable in question.

Substituting the coordinates above and rearranging the equation gives

Motion of the double compound pendulum (from numerical integration of the equations of motion)
Long exposure of double pendulum exhibiting chaotic motion (tracked with an LED)

There is only one conserved quantity (the energy), and no conserved momenta. The two momenta may be written as

and

These expressions may be inverted to get

and

The remaining equations of motion are written as

and

These last four equations are explicit formulae for the time evolution of the system given its current state. It is not possible to go further and integrate these equations analyticallyPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park., to get formulae for θ1 and θ2 as functions of time. It is however possible to perform this integration numerically using the Runge Kutta method or similar techniques.

Chaotic motion

Graph of the time for the pendulum to flip over as a function of initial conditions

The double pendulum undergoes chaotic motion, and shows a sensitive dependence on initial conditions. The image to the right shows the amount of elapsed time before the pendulum "flips over," as a function of initial conditions. Here, the initial value of θ1 ranges along the x-direction, from −3 to 3. The initial value θ2 ranges along the y-direction, from −3 to 3. The colour of each pixel indicates whether either pendulum flips within (green), within (red), (purple) or (blue). Initial conditions that don't lead to a flip within are plotted white.

The boundary of the central white region is defined in part by energy conservation with the following curve:

Within the region defined by this curve, that is if

then it is energetically impossible for either pendulum to flip. Outside this region, the pendulum can flip, but it is a complex question to determine when it will flip. Similar behavior is observed for a double pendulum composed of two point masses rather than two rods with distributed mass.[2]

The lack of a natural excitation frequency has led to the use of double pendulum systems in seismic resistance designs in buildings, where the building itself is the primary inverted pendulum, and a secondary mass is connected to complete the double pendulum.

See also

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

External links

Template:Chaos theory

  1. Levien RB and Tan SM. Double Pendulum: An experiment in chaos.American Journal of Physics 1993; 61 (11): 1038
  2. Alex Small, Sample Final Project: One Signature of Chaos in the Double Pendulum, (2013). A report produced as an example for students. Includes a derivation of the equations of motion, and a comparison between the double pendulum with 2 point masses and the double pendulum with 2 rods.