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Kapitza's pendulum
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<h2 id="notation">Notation</h2> <p>Denote the vertical axis as y {\displaystyle y} and the horizontal axis as x {\displaystyle x} so that the motion of pendulum happens in the ( x {\displaystyle x} - y {\displaystyle y} ) plane. The following notation will be used </p> <ul><li> ν {\displaystyle \nu } —frequency of the vertical oscillations of the suspension,</li> <li> a {\displaystyle a} — amplitude of the oscillations of the suspension,</li> <li> ω 0 = g / l {\displaystyle \omega _{0}={\sqrt {g/l}}} — proper frequency of the mathematical pendulum,</li> <li> g {\displaystyle g} — free fall acceleration,</li> <li> l {\displaystyle l} — length of rigid and light pendulum,</li> <li> m {\displaystyle m} — mass.</li></ul> <p>Denoting the angle between pendulum and downward direction as φ {\displaystyle \varphi } the time dependence of the position of pendulum gets written as </p> { x = l sin φ y = − l cos φ − a cos ν t {\displaystyle {\begin{cases}x&=l\sin \varphi \\y&=-l\cos \varphi -a\cos \nu t\end{cases}}} <h2 id="energy">Energy</h2> <p>The <a href="/facts/Potential_energy/I03KLbmn">potential energy</a> of the pendulum is due to gravity and is defined by, in terms of the vertical position, as </p> E P O T = − m g ( l cos φ + a cos ν t ) . {\displaystyle E_{\mathrm {POT} }=-mg(l\cos \varphi +a\cos \nu t).\,} <p>The <a href="/facts/Kinetic_energy/aJRxUd4p">kinetic energy</a> in addition to the standard term E K I N = m l 2 φ ˙ 2 / 2 {\displaystyle E_{\mathrm {KIN} }=ml^{2}{\dot {\varphi }}^{2}/2} , describing velocity of a mathematical pendulum, there is a contribution due to vibrations of the suspension </p> E K I N = m l 2 2 φ ˙ 2 + m a l ν sin ( ν t ) sin ( φ ) φ ˙ + m a 2 ν 2 2 sin 2 ( ν t ) . {\displaystyle E_{\mathrm {KIN} }={\frac {ml^{2}}{2}}{\dot {\varphi }}^{2}+mal\nu ~\sin(\nu t)\sin(\varphi )~{\dot {\varphi }}+{\frac {ma^{2}\nu ^{2}}{2}}\sin ^{2}(\nu t)\;.} <p>The total energy is given by the sum of the kinetic and potential energies E = E K I N + E P O T {\displaystyle E=E_{\mathrm {KIN} }+E_{\mathrm {POT} }} and the <a href="/facts/Lagrangian_mechanics/sTxyqWz8">Lagrangian</a> by their difference L = E K I N − E P O T {\displaystyle L=E_{\mathrm {KIN} }-E_{\mathrm {POT} }} . </p><p>The total energy is conserved in a mathematical pendulum, so time t {\displaystyle t} dependence of the potential E P O T {\displaystyle E_{\mathrm {POT} }} and kinetic E K I N {\displaystyle E_{\mathrm {KIN} }} energies is symmetric with respect to the horizontal line. According to the <a href="/facts/Virial_theorem/w3qwDrkr">virial theorem</a> 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. </p><p>In the case of vibrating suspension the system is no longer a <a href="/facts/Closed_system_(thermodynamics)/8KEjjgQE">closed one</a> and the total energy is no longer conserved. The kinetic energy is more sensitive to vibration compared to the potential one. The potential energy E P O T = m g y {\displaystyle E_{\mathrm {POT} }=mgy} is bound from below and above − m g ( l + a ) < E P O T < m g ( l + a ) {\displaystyle -mg(l+a)<E_{\mathrm {POT} }<mg(l+a)} while the kinetic energy is bound only from below E K I N ≥ 0 {\displaystyle E_{\mathrm {KIN} }\geq 0} . For high frequency of vibrations ν {\displaystyle \nu } the kinetic energy can be large compared to the potential energy. </p> <h2 id="equations-of-motion">Equations of motion</h2> <p>Motion of pendulum satisfies <a href="/facts/Euler%25E2%2580%2593Lagrange_equation/Ymq3r3Dh">Euler–Lagrange equations</a>. The dependence of the phase φ {\displaystyle \varphi } of the pendulum on its position satisfies the equation:<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> d d t ∂ L ∂ φ ˙ = ∂ L ∂ φ , {\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {\varphi }}}}={\frac {\partial L}{\partial \varphi }},} <p>where the Lagrangian L {\displaystyle L} reads </p> L = m l 2 2 φ ˙ 2 + m l ( g + a ν 2 cos ν t ) cos φ , {\displaystyle L={\frac {ml^{2}}{2}}{\dot {\varphi }}^{2}+ml(g+a~\nu ^{2}\cos \nu t)\cos \varphi ,} <p>up to irrelevant total time derivative terms. The differential equation </p> φ ¨ = − ( g + a ν 2 cos ν t ) sin φ l , {\displaystyle {\ddot {\varphi }}=-(g+a~\nu ^{2}\cos \nu t){\frac {\sin \varphi }{l}},} <p>which describes the movement of the pendulum is nonlinear due to the sin φ {\displaystyle \sin \varphi } factor. </p> <h2 id="equilibrium-positions">Equilibrium positions</h2> <p>Kapitza's pendulum model is more general than the <a href="/facts/Pendulum_(mathematics)/e7ldI0ot">simple pendulum</a>. The Kapitza model reduces to the latter in the limit a = 0 {\displaystyle a=0} . In that limit, the tip of the pendulum describes a circle: x 2 + y 2 = l 2 = constant {\displaystyle x^{2}+y^{2}=l^{2}={\text{constant}}} . If the energy in the initial moment is larger than the maximum of the potential energy E > m g l {\displaystyle E>mgl} then the trajectory will be closed and cyclic. If the initial energy is smaller E < m g l {\displaystyle E<mgl} then the pendulum will oscillate close to the only stable point ( x , y ) = ( 0 , − l ) {\displaystyle (x,y)=(0,-l)} . </p><p>When the suspension is vibrating with a small amplitude a ≪ l {\displaystyle a\ll l} and with a frequency ν ≫ ω 0 {\displaystyle \nu \gg \omega _{0}} much higher than the proper frequency ω 0 {\displaystyle \omega _{0}} , the angle φ {\displaystyle \varphi } may be viewed as a superposition φ = φ 0 + ξ {\displaystyle \varphi =\varphi _{0}+\xi } of a "slow" component φ 0 {\displaystyle \varphi _{0}} and a rapid oscillation ξ {\displaystyle \xi } with small amplitude due to the small but rapid vibrations of the suspension. Technically, we perform a <a href="/facts/Perturbative/lYGgk9Ph">perturbative</a> expansion in the "<a href="/facts/Coupling_constant/Queop1TR">coupling constants</a>" ( a / l ) , ( ω 0 / ν ) ≪ 1 {\displaystyle (a/l),(\omega _{0}/\nu )\ll 1} while treating the ratio ( a / l ) ( ν / ω 0 ) {\displaystyle (a/l)(\nu /\omega _{0})} as fixed. The perturbative treatment becomes exact in the <a href="/facts/Double_scaling_limit/r7Cgrom0">double scaling limit</a> a → 0 , ν → ∞ {\displaystyle a\to 0,\nu \to \infty } . More precisely, the rapid oscillation ξ {\displaystyle \xi } is defined as </p> ξ = a l sin φ 0 cos ν t . {\displaystyle \xi ={\frac {a}{l}}\sin \varphi _{0}~\cos \nu t.\,} <p>The equation of motion for the "slow" component φ 0 {\displaystyle \varphi _{0}} becomes </p> φ ¨ 0 = φ ¨ − ξ ¨ = − ( g + a ν 2 cos ν t ) sin φ l − a l ( φ ¨ 0 cos φ 0 cos ν t − φ ˙ 0 2 sin φ 0 cos ν t − 2 ν φ ˙ 0 cos φ 0 sin ν t − ν 2 sin φ 0 cos ν t ) = − g l sin φ 0 − ( g + a ν 2 cos ν t ) 1 l ( ξ cos φ 0 + O ( ξ 2 ) ) − a l ( φ ¨ 0 cos φ 0 cos ν t − φ ˙ 0 2 sin φ 0 cos ν t − 2 ν φ ˙ 0 cos φ 0 sin ν t ) . {\displaystyle {\begin{aligned}{\ddot {\varphi }}_{0}={\ddot {\varphi }}-{\ddot {\xi }}&=-(g+a~\nu ^{2}\cos \nu t){\frac {\sin \varphi }{l}}\\&{}\quad {}-{\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)\\[8pt]&=-{\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)\\&{}\quad {}-{\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).\end{aligned}}} <p>Time-averaging over the rapid ν {\displaystyle \nu } -oscillation yields to leading order </p> φ ¨ 0 = − g l sin φ 0 − 1 2 ( a ν l ) 2 sin φ 0 cos φ 0 . {\displaystyle {\ddot {\varphi }}_{0}=-{\frac {g}{l}}\sin \varphi _{0}-{\frac {1}{2}}\left({\frac {a\nu }{l}}\right)^{2}\sin \varphi _{0}\cos \varphi _{0}.} <p>The "slow" equation of motion becomes </p> m l 2 φ ¨ 0 = − ∂ V e f f ∂ φ 0 , {\displaystyle ml^{2}{\ddot {\varphi }}_{0}=-{\frac {\partial V_{\mathrm {eff} }}{\partial \varphi _{0}}}\;,} <p>by introducing an <a href="/facts/Effective_potential/qCWSYXVx">effective potential</a> </p> V e f f = − m g l cos φ 0 + m ( a ν 2 sin φ 0 ) 2 . {\displaystyle V_{\mathrm {eff} }=-mgl\cos \varphi _{0}+m\left({\frac {a\nu }{2}}\sin \varphi _{0}\right)^{2}.} <p>It turns out<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> that the effective potential V e f f {\displaystyle V_{\mathrm {eff} }} has two minima if ( a ν ) 2 > 2 g l {\displaystyle (a\nu )^{2}>2gl} , or equivalently, ( a / l ) ( ν / ω 0 ) > 2 {\displaystyle (a/l)(\nu /\omega _{0})>{\sqrt {2}}} . The first minimum is in the same position ( x , y ) = ( 0 , − l ) {\displaystyle (x,y)=(0,-l)} as the mathematical pendulum and the other minimum is in the upper vertical position ( x , y ) = ( 0 , l ) {\displaystyle (x,y)=(0,l)} . As a result the upper vertical position, which is unstable in a mathematical pendulum, can become stable in Kapitza's pendulum. </p> <h2 id="bifurcation-and-chaos">Bifurcation and chaos</h2> <p>For fixed forcing frequency ν {\displaystyle \nu } , and increasing forcing amplitude a {\displaystyle a} , the inverted point φ = π {\displaystyle \varphi =\pi } becomes a stable equilibrium, and then it would become unstable again. The point undergoes a <a href="/facts/Hopf_bifurcation/WuFmnr96">Hopf bifurcation</a>: the stable equilibrium expands into a stable oscillation around φ = π {\displaystyle \varphi =\pi } , at frequency ν / 2 {\displaystyle \nu /2} , and amplitude ∝ a − a c {\displaystyle \propto {\sqrt {a-a_{c}}}} where a c {\displaystyle a_{c}} is the critical amplitude.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p><p>For even larger amplitude, the stable oscillation becomes unstable, and the pendulum would start rotating. At even larger amplitudes, the rotating mode is also destroyed, and a <a href="/facts/Strange_attractor/FAx5Db4w">strange attractor</a> appears by <a href="/facts/Period-doubling_bifurcation/UchwCLYh">period-doubling cascade</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> <h2 id="rotating-solutions">Rotating solutions</h2> <p>The rotating solutions of the Kapitza's pendulum occur when the pendulum rotates around the pivot point at the same frequency that the pivot point is driven. There are two rotating solutions, one for a rotation in each direction. We shift to the rotating reference frame using φ → φ ′ ± ν t {\displaystyle \varphi \rightarrow \varphi ^{\prime }\pm \nu t} and the equation for φ {\displaystyle \varphi } becomes: </p> φ ¨ ′ = − 1 l [ 1 2 a ν 2 sin ( φ ′ ) + g sin ( φ ′ ± ν t ) + 1 2 a ν 2 sin ( φ ′ ± 2 ν t ) ] . {\displaystyle {\ddot {\varphi }}^{\prime }=-{\frac {1}{l}}\left[{\frac {1}{2}}a\nu ^{2}\sin(\varphi ^{\prime })+g\sin(\varphi ^{\prime }\pm \nu t)+{\frac {1}{2}}a\nu ^{2}\sin(\varphi ^{\prime }\pm 2\nu t)\right]\;.} <p>Again considering the limit in which ν {\displaystyle \nu } is much higher than the proper frequency ω 0 {\displaystyle \omega _{0}} , we find that the rapid- ν {\displaystyle \nu } slow- φ 0 ′ {\displaystyle \varphi _{0}^{\prime }} limit leads to the equation: </p> φ ¨ 0 ′ = − 1 2 l a ν 2 sin φ 0 ′ . {\displaystyle {\ddot {\varphi }}_{0}^{\prime }=-{\frac {1}{2l}}a\nu ^{2}\sin \varphi _{0}^{\prime }\;.} <p>The effective potential is just that of a simple pendulum equation. There is a stable equilibrium at φ 0 ′ = 0 {\displaystyle \varphi _{0}^{\prime }=0} and an unstable equilibrium at φ 0 ′ = π {\displaystyle \varphi _{0}^{\prime }=\pi } . </p> <h2 id="phase-portrait">Phase portrait</h2> <p>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 a ≈ l {\displaystyle a\approx l} .<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> Increasing the amplitude of driving oscillations to half of the pendulum length a = l / 2 {\displaystyle a=l/2} leads to the phase portrait shown in the figure. </p><p>Further increase of the amplitude to a ≈ l {\displaystyle a\approx l} 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 a {\displaystyle a} . </p> <h2 id="interesting-facts">Interesting facts</h2> <ul><li>Kapitza noted that a <a href="/facts/Pendulum_clock/R6uQYRNw">pendulum clock</a> with a vibrating pendulum suspension always goes faster than a clock with a fixed suspension.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a></li> <li><a href="/facts/Walking/Mow1X5Im">Walking</a> 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.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a></li></ul> <h2 id="literature">Literature</h2> <h2 id="external-links">External links</h2> <ul><li>Demonstration video at <a href="https://www.youtube.com/watch?v=cjGqxF79ITI">Kapitza's Pendulum - YouTube</a></li> <li><a href="http://demonstrations.wolfram.com/KapitzasPendulum/">Interactive demonstration</a> at <a href="/facts/Wolfram_Demonstrations_Project/l0sh4gln">Wolfram Demonstrations Project</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Kapitza P. L. (1951). "Dynamic stability of a pendulum when its point of suspension vibrates". Soviet Phys. JETP. 21: 588–597.; Kapitza P. L. (1951). "Pendulum with a vibrating suspension". Usp. Fiz. Nauk. 44: 7–15. doi:10.3367/UFNr.0044.195105b.0007. <a href="https://doi.org/10.3367%2FUFNr.0044.195105b.0007" target="_blank">https://doi.org/10.3367%2FUFNr.0044.195105b.0007</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Stephenson Andrew (1908). "XX.On induced stability". Philosophical Magazine. 6. 15 (86): 233–236. doi:10.1080/14786440809463763. <a href="https://zenodo.org/record/1430846" target="_blank">https://zenodo.org/record/1430846</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Kapitza P. L. (1951). "Dynamic stability of a pendulum when its point of suspension vibrates". Soviet Phys. JETP. 21: 588–597.; Kapitza P. L. (1951). "Pendulum with a vibrating suspension". Usp. Fiz. Nauk. 44: 7–15. doi:10.3367/UFNr.0044.195105b.0007. <a href="https://doi.org/10.3367%2FUFNr.0044.195105b.0007" target="_blank">https://doi.org/10.3367%2FUFNr.0044.195105b.0007</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>L. D. Landau, E. M. Lifshitz (1960). Mechanics. Vol. 1 (1st ed.). Pergamon Press. ASIN B0006AWV88. <a href="/wiki/Lev_Landau" target="_blank">/wiki/Lev_Landau</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Бутиков Е. И. «Маятник с осциллирующим подвесом (к 60-летию маятника Капицы»), учебное пособие. <a href="http://faculty.ifmo.ru/butikov/Russian/ParamPendulum.pdf" target="_blank">http://faculty.ifmo.ru/butikov/Russian/ParamPendulum.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Blackburn, James A.; Smith, H. J. T.; Gro/nbech-Jensen, N. (October 1992). "Stability and Hopf bifurcations in an inverted pendulum". American Journal of Physics. 60 (10): 903–908. Bibcode:1992AmJPh..60..903B. doi:10.1119/1.17011. ISSN 0002-9505. <a href="https://pubs.aip.org/aapt/ajp/article/60/10/903-908/1054037" target="_blank">https://pubs.aip.org/aapt/ajp/article/60/10/903-908/1054037</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>V. P. Krainov (2002). Selected Mathematical Methods in Theoretical Physics (1st ed.). Taylor & Francis. ISBN 978-0-415-27234-6. <a href="978-0-415-27234-6" target="_blank">978-0-415-27234-6</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Kapitza P. L. (1951). "Dynamic stability of a pendulum when its point of suspension vibrates". Soviet Phys. JETP. 21: 588–597.; Kapitza P. L. (1951). "Pendulum with a vibrating suspension". Usp. Fiz. Nauk. 44: 7–15. doi:10.3367/UFNr.0044.195105b.0007. <a href="https://doi.org/10.3367%2FUFNr.0044.195105b.0007" target="_blank">https://doi.org/10.3367%2FUFNr.0044.195105b.0007</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Blackburn, James A.; Smith, H. J. T.; Gro/nbech-Jensen, N. (October 1992). "Stability and Hopf bifurcations in an inverted pendulum". American Journal of Physics. 60 (10): 903–908. Bibcode:1992AmJPh..60..903B. doi:10.1119/1.17011. ISSN 0002-9505. <a href="https://pubs.aip.org/aapt/ajp/article/60/10/903-908/1054037" target="_blank">https://pubs.aip.org/aapt/ajp/article/60/10/903-908/1054037</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>McLaughlin, John B. (February 1981). "Period-doubling bifurcations and chaotic motion for a parametrically forced pendulum". Journal of Statistical Physics. 24 (2): 375–388. Bibcode:1981JSP....24..375M. doi:10.1007/BF01013307. ISSN 0022-4715. S2CID 55706875. <a href="http://link.springer.com/10.1007/BF01013307" target="_blank">http://link.springer.com/10.1007/BF01013307</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Koch, B. P.; Leven, R. W.; Pompe, B.; Wilke, C. (1983-07-04). "Experimental evidence for chaotic behaviour of a parametrically forced pendulum". Physics Letters A. 96 (5): 219–224. Bibcode:1983PhLA...96..219K. doi:10.1016/0375-9601(83)90336-5. ISSN 0375-9601. <a href="https://dx.doi.org/10.1016/0375-9601%2883%2990336-5" target="_blank">https://dx.doi.org/10.1016/0375-9601%2883%2990336-5</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>G. E. Astrakharchik, N. A. Astrakharchik «Numerical study of Kapitza pendulum» arXiv:1103.5981 (2011) <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Time motion of Kapitza's pendulum can be modeled in online java applets on the following sites: "Inverted Vibrating Pendulum Physics Simulation". Archived from the original on 2011-10-01. Retrieved 2011-04-08. Arbitrary parameters of the system can be used and can be inserted manually <a href="https://web.archive.org/web/20111001191853/http://www.myphysicslab.com/beta/Inverted-pendulum.html" target="_blank">https://web.archive.org/web/20111001191853/http://www.myphysicslab.com/beta/Inverted-pendulum.html</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Butikov, Eugene I. "Kapitza Pendulum: A Physically Transparent Simple Explanation" (PDF). p. 8. Retrieved September 1, 2020. <a href="http://butikov.faculty.ifmo.ru/InvPendulumCNS.pdf" target="_blank">http://butikov.faculty.ifmo.ru/InvPendulumCNS.pdf</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Quintanilla, José; Perez, Moises; Balderas, Rodolfo; González, Alejandro; Cardenas, Antonio; Maya, Mauro; Piovesan, Davide (May 2021). "Inertial Stabilization of Upright Posture while walking". 2021 10th International IEEE/EMBS Conference on Neural Engineering (NER). pp. 849–852. doi:10.1109/NER49283.2021.9441114. ISBN 978-1-7281-4337-8. S2CID 235307555. <a href="978-1-7281-4337-8" target="_blank">978-1-7281-4337-8</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> </ol>