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| [[File:Kapitza pendulum.svg|thumb|Drawing showing how a Kapitza pendulum can be constructed: a motor rotates a crank at a high speed, the crank vibrates a lever arm up and down which the pendulum is attached to with a pivot.]]
| | Next - GEN Gallery is a full incorporated Image Gallery plugin for Word - Press which has a Flash slideshow option. Affilo - Theme is the guaranteed mixing of wordpress theme that Mark Ling use for his internet marketing career. PSD files are incompatible to browsers and are suppose to be converted into wordpress compatible files so that it opens up in browser. After confirming the account, login with your username and password at Ad - Mob. The top 4 reasons to use Business Word - Press Themes for a business website are:. <br><br>purcase and download - WPZOOM Tribune wordpress Theme, find and use the WPZOOM Discount Code. You do not catch a user's attention through big and large pictures that usually takes a millennium to load up. Well Managed Administration The Word - Press can easily absorb the high numbers of traffic by controlling the server load to make sure that the site works properly. Apart from these, you are also required to give some backlinks on other sites as well. But in case you want some theme or plugin in sync with your business needs, it is advisable that you must seek some professional help. <br><br>Here's more on [http://www.bentleygsa.org/cssa/bbs/home/link.php?url=https://wordpress.org/plugins/ready-backup/ backup plugin] take a look at the internet site. Usually, Wordpress owners selling the ad space on monthly basis and this means a residual income source. As of now, Pin2Press is getting ready to hit the market. This platform can be customizedaccording to the requirements of the business. Newer programs allow website owners and internet marketers to automatically and dynamically change words in their content to match the keywords entered by their web visitors in their search queries'a feat that they cannot easily achieve with older software. Search engine optimization pleasant picture and solution links suggest you will have a much better adjust at gaining considerable natural site visitors. <br><br>Digg Digg Social Sharing - This plugin that is accountable for the floating social icon located at the left aspect corner of just about every submit. * Robust CRM to control and connect with your subscribers. A higher percentage of women are marrying at older ages,many are delaying childbearing until their careers are established, the divorce rate is high and many couples remarry and desire their own children. IVF ,fertility,infertility expert,surrogacy specialist in India at Rotundaivf. It does take time to come up having a website that gives you the much needed results hence the web developer must be ready to help you along the route. <br><br>As a open source platform Wordpress offers distinctive ready to use themes for free along with custom theme support and easy customization. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. In simple words, this step can be interpreted as the planning phase of entire PSD to wordpress conversion process. Extra investment in Drupal must be bypassed as value for money is what Drupal provides. Customers within a few seconds after visiting a site form their opinion about the site. |
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| '''Kapitza's pendulum''' is a rigid [[pendulum]] in which the pivot point vibrates in a vertical direction, up and down. It is named after Russian [[Nobel prize|Nobel laureate]] physicist [[Pyotr Kapitza]], who in 1951 developed a theory which successfully explains some of its unusual properties.<ref name="Kapitza">P.L. Kapitza, “Dynamic stability of a pendulum when its point of suspension vibrates”, Soviet Phys. JETP 21, 588–592 (1951); P.L. Kapitza, “Pendulum with a vibrating suspension” Usp. Fiz. Nauk, 44, 7-15 (1951).</ref> The unique feature of the Kapitza pendulum is that the vibrating suspension can cause it to balance stably in an inverted position, with the bob above the suspension point. In the usual [[Pendulum (mathematics)|pendulum]] with a fixed suspension, the only stable equilibrium position is with the bob hanging below the suspension point; the inverted position is a point of [[Mechanical equilibrium|unstable equilibrium]], and the smallest perturbation moves the pendulum out of equilibrium. In [[nonlinear control theory]] the Kapitza pendulum is used as an example of a [[parametric oscillator]] that demonstrates the concept of "dynamic stabilization".
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| The pendulum was first described by A. Stephenson in 1908, who found that the upper vertical position of the pendulum might be stable when the driving frequency is fast<ref>A. Stephenson "On an induced stability" Phil. Mag. 15, 233 (1908). [http://www.tandfonline.com/doi/abs/10.1080/14786440809463763]</ref> Yet until the 1950s there was no explanation for this highly unusual and counterintuitive phenomenon. Pyotr Kapitza was the first to analyze it in 1951.<ref name="Kapitza" /> He carried out a number of experimental studies and as well provided an analytical insight into the reasons of stability by splitting the motion into "fast" and "slow" variables and by introducing an effective potential. This innovative work created a new subject in physics, that is vibrational mechanics. Kapitza's method is used for description of periodic processes in [[atomic physics]], [[plasma physics]], [[cybernetical physics]]. The effective potential which describes the "slow" component of motion is described in "Mechanics" volume of the [[Lev Landau|Landau]]'s [[Course of Theoretical Physics]].<ref>{{cite book
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| |author=[[Lev Landau|L.D. Landau]], [[Evgeny Lifshitz|E.M. Lifshitz]]
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| |year=1960
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| |title=Mechanics
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| |edition=1st |volume=Vol. 1
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| |publisher=[[Pergamon Press]]
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| |isbn=
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| |oclc=
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| |asin=B0006AWV88
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| }}</ref>
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| Another interesting feature of the Kapitza pendulum system is that the bottom equilibrium position, with the pendulum hanging down below the pivot, is no longer stable. Any tiny deviation from the vertical increases in amplitude with time.<ref>Бутиков Е. И. «Маятник с осциллирующим подвесом (к 60-летию маятника Капицы»), [http://faculty.ifmo.ru/butikov/Russian/ParamPendulum.pdf учебное пособие].</ref> Also [[parametric resonance]] can occur in this position, and [[Chaos theory|chaotic regimes]] can be realized in that system when [[strange attractor]]s are present in the [[Poincaré section]] .
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| == Notation ==
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| [[File:KapitzaPendulumScheme.gif|thumb|240px|right|Kapitza's pendulum scheme]]
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| Denote the vertical axis as <math>y</math> and the horizontal axis as <math>x</math> so that the motion of pendulum happens in the (<math>x</math>—<math>y</math>) plane. The following notation will be used
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| * <math>\nu</math>—frequency of the vertical oscillations of the suspension,
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| * <math>a</math> — amplitude of the oscillations of the suspension,
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| * <math>\omega_0 = \sqrt{g/l}</math> — proper frequency of the mathematical pendulum,
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| * <math>g</math> — free fall acceleration,
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| * <math>l</math> — length of rigid and light pendulum,
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| * <math>m</math> — mass.
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| Denoting the angle between pendulum and downward direction as <math>\varphi</math> the time dependence of the position of pendulum gets written as
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| <center><math>\begin{matrix}
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| \left\{
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| \begin{matrix}
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| x &=& l \sin \varphi\\
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| y &=& - l \cos \varphi - a \cos \nu t
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| \end{matrix} \right.
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| \end{matrix}</math></center>
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| == Energy ==
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| The [[potential energy]] of the pendulum is due to gravity and is defined by of the vertical position as
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| <center><math>\begin{matrix}
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| E_{POT} = - m g (l \cos \varphi + a \cos \nu t)\;.
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| \end{matrix}</math></center>
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| The [[kinetic energy]] in addition to the standard term <math>E_{KIN}=m l^2 \dot \varphi^2 /2</math>, describing velocity of a mathematical pendulum, there is a contribution due to vibrations of the suspension
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| <center><math>\begin{matrix}
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| E_{KIN}
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| = \frac{m l^2 }{2} \dot \varphi^2 + m a l \nu ~\sin(\nu t) \sin(\varphi)~\dot\varphi + \frac{m a^2 \nu^2}{2} \sin^2(\nu t)\;.
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| \end{matrix}</math></center>
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| The total energy is given by the sum of the kinetic and potential energies <math>E = E_{KIN} + E_{POT}</math> and the [[Lagrangian]] by their difference <math>L = E_{KIN} - E_{POT}</math>.
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| The total energy is conserved in a mathematical pendulum, so time <math>t</math> dependence of the potential <math>E_{POT}</math> and kinetic <math>E_{KIN}</math> energies is symmetric with respect to the horizontal line. According to the [[virial theorem]] the mean kinetic and potential energies in harmonic oscillator are equal. This means that the line of symmetry corresponds to half of the total energy.
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| In the case of vibrating suspension the system is no longer is a [[Closed system (thermodynamics)|closed one]] and the total energy is no longer conserved. The kinetic energy is more sensitive to vibration compared to the potential one. The potential energy <math>E_{POT} = mgy</math> is bound from below and above <math>-mg(l+a)<E_{POT}<mg(l+a)</math> while the kinetic energy is bound only from below <math>E_{KIN}\ge 0</math>. For high frequency of vibrations <math>\nu</math> the kinetic energy can be large compared to the potential energy.
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| == Equations of motion ==
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| Motion of pendulum satisfies [[Euler-Lagrange equation]]s. The dependence of the phase <math>\varphi</math> of the pendulum on its position satisfies the equation:<ref>{{cite book
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| |author=V.P. Krainov
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| |year=2002
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| |title=Selected Mathematical Methods in Theoretical Physics
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| |edition=1st
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| |publisher=[[Taylor & Francis]]
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| |isbn=978-0-415-27234-6
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| |oclc=
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| }}</ref>
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| <center><math>\begin{matrix}
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| \frac{d}{dt} \frac{\partial L}{\partial \dot \varphi} = \frac{\partial L}{\partial \varphi}\;,
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| \end{matrix}</math></center>
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| where the Lagrangian <math>L</math> reads
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| <center><math>\begin{matrix}
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| L = \frac{m l^2 }{2} \dot \varphi^2 + ml( g + a~\nu^2\cos\nu t) \cos \varphi\;,
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| \end{matrix}</math></center>
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| up to irrelevant total time derivative terms. The differential equation
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| <center><math>\begin{matrix}
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| \ddot \varphi = - (g+a~\nu^2\cos\nu t) \frac{\sin \varphi}{l}\;,
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| \end{matrix}</math></center>
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| which describes the movement of the pendulum is nonlinear due to the <math>\sin\varphi</math> factor. The presence of the nonlinear term might lead to [[chaos theory|chaotic]] motion and to appearance of [[strange attractor]]s.
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| == Equilibrium positions ==
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| Model of Kapitza's pendulum is more general than the model of mathematical pendulum. The latter is reproduced in the limit <math>a = 0</math>. Its [[phase portrait]] is a simple circle <math>x^2+y^2 = l^2 = const</math>. If the energy in the initial moment was larger than the maximum of the potential energy <math>E > mgl</math> then the trajectory will be closed and cyclic. If the initial energy is smaller <math>E < mgl</math> then the pendulum will oscillate close to the only stably point <math>(x,y) = (0,-l)</math>.
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| When the suspension is vibrating with a small amplitude <math> a \ll l </math> and with a frequency <math>\nu\gg \omega_0 </math> much higher than the proper frequency <math>\omega_0</math>, the angle <math> \varphi</math> may be viewed as a superposition <math> \varphi=\varphi_0+\xi</math> of a "slow" component <math>\varphi_0</math> and a rapid oscillation <math>\xi</math> with small amplitude due to the small but rapid vibrations of the suspension. Technically, we perform a [[perturbative]] expansion in the "[[coupling constant]]s" <math> (a/l),(\omega_0/\nu) \ll 1 </math> while treating the ratio <math> (a/l)(\nu/\omega_0)</math> as fixed. The perturbative treatment becomes exact in the [[double scaling limit]] <math> a \to 0 ,\nu\to \infty</math>. More precisely, the rapid oscillation <math>\xi</math> is defined as
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| <center><math>\begin{matrix}
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| \xi = \frac{a}{l}\sin\varphi_0 ~\cos\nu t\;.
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| \end{matrix}</math></center>
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| The equation of motion for the "slow" component <math>\varphi_0</math> becomes
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| <center><math>\begin{array}{rcl}
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| \ddot\varphi_0 = \ddot\varphi - \ddot\xi &=& -(g+a~\nu^2\cos\nu t)\frac{\sin\varphi}{l} - \frac{a}{l}\left(\ddot\varphi_0 \cos \varphi_0 ~\cos\nu t -\dot\varphi_0^2\sin\varphi_0 ~\cos\nu t - 2\nu\dot\varphi_0\cos\varphi_0~\sin\nu t - \nu^2\sin \varphi_0 ~\cos\nu t \right) \\
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| &=& -\frac{g}{l}\sin\varphi_0 -(g+a~\nu^2\cos\nu t) \frac{1}{l}\left(\xi\cos\varphi_0 + O(\xi^2)\right) - \frac{a}{l}\left( \ddot\varphi_0 \cos\varphi_0 ~\cos\nu t -\dot\varphi_0^2\sin\varphi_0 ~\cos\nu t - 2\nu\dot\varphi_0\cos\varphi_0~\sin\nu t \right)\;.
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| \end{array}</math></center>
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| Time averaging over the rapid <math>\nu</math>-oscillation yields to leading order
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| <center><math>\begin{matrix}
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| \ddot \varphi_0 = - \frac{g}{l}\sin \varphi_0 - \frac{1}{2}(\frac{a\nu}{l})^2\sin \varphi_0 \cos \varphi_0 \;.
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| \end{matrix}</math></center>
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| The "slow" equation of motion becomes
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| <center><math>\begin{matrix}
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| m l^2\ddot \varphi_0 = -\frac{\partial V_{\mathrm{eff}}}{\partial \varphi_0} \;,
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| \end{matrix}</math></center>
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| by introducing an [[effective potential]]
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| <center><math>\begin{matrix}
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| V_{\mathrm{eff}} = - mgl \cos \varphi_0 + m (\frac{a\nu}{2}\sin \varphi_0)^2 \;.
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| \end{matrix}</math></center>
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| It turns out<ref name="Kapitza" /> that the effective potential <math> V_{\mathrm{eff}} </math> has two minima if <math>(a\nu)^2 > 2gl </math>, or equivalently, <math> (a/l)(\nu/\omega_0) > \sqrt{2} </math>. The first minimum is in the same position <math>(x,y)=(0,-l)</math> as the mathematical pendulum and the other minimum is in the upper vertical position <math>(x,y)=(0,l)</math>. As a result the upper vertical position, which is unstable in a mathematical pendulum, can become stable in Kapitza's pendulum.
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| == Phase portrait ==
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| Interesting phase portraits might be obtained in regimes which are not accessible within analytic descriptions, for example in the case of large amplitude of the suspension <math>a \approx l</math>.<ref>G.E. Astrakharchik, N.A. Astrakharchik «Numerical study of Kapitza pendulum» [http://arxiv.org/pdf/1103.5981v1 arXiv:1103.5981 (2011)]</ref><ref>
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| Time motion of Kapitza’s pendulum can be modeled in online java applets he sites
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| http://www.myphysicslab.com/beta/Inverted-pendulum.html
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| http://faculty.ifmo.ru/butikov/Nonlinear/index.html
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| Arbitrary parameters of the system can be used and can be inserted manually.</ref> Increasing the amplitude of driving oscillations to half of the pendulum length <math>a = l/2</math> leads to the phase portrait shown in the figure.
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| Further increase of the amplitude to <math>a\approx l</math>) leads to full filling of the internal points of the phase space, if before some points of the phase space were not accessible, now system can reach any of the internal points. This situation holds also for larger values of <math>a</math>.
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| == Interesting facts ==
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| * Kapitza noted that a [[pendulum clock]] with a vibrating pendulum suspension always goes faster than a clock with a fixed suspension.
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| * [[Walking]] is defined by an 'inverted pendulum' gait in which the body vaults over the stiff limb or limbs with each step. Increased stability during walking might be related to stability of Kapitza's pendulum. This applies regardless of the number of limbs - even arthropods with six, eight or more limbs.
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| == Literature ==
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| {{Reflist}}
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| {{DEFAULTSORT:Kapitza's Pendulum}}
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| [[Category:Pendulums]]
| |
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