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Spherical pendulum
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<h2 id="lagrangian-mechanics">Lagrangian mechanics</h2> <p class="note">Main article: <a href="/facts/Lagrangian_mechanics/sTxyqWz8">Lagrangian mechanics</a></p> <p>Routinely, in order to write down the kinetic T = 1 2 m v 2 {\displaystyle T={\tfrac {1}{2}}mv^{2}} and potential V {\displaystyle V} parts of the Lagrangian L = T − V {\displaystyle L=T-V} in arbitrary generalized coordinates the position of the mass is expressed along Cartesian axes. Here, following the conventions shown in the diagram, </p> x = l sin θ cos ϕ {\displaystyle x=l\sin \theta \cos \phi } y = l sin θ sin ϕ {\displaystyle y=l\sin \theta \sin \phi } z = l ( 1 − cos θ ) {\displaystyle z=l(1-\cos \theta )} . <p>Next, time derivatives of these coordinates are taken, to obtain velocities along the axes </p> x ˙ = l cos θ cos ϕ θ ˙ − l sin θ sin ϕ ϕ ˙ {\displaystyle {\dot {x}}=l\cos \theta \cos \phi \,{\dot {\theta }}-l\sin \theta \sin \phi \,{\dot {\phi }}} y ˙ = l cos θ sin ϕ θ ˙ + l sin θ cos ϕ ϕ ˙ {\displaystyle {\dot {y}}=l\cos \theta \sin \phi \,{\dot {\theta }}+l\sin \theta \cos \phi \,{\dot {\phi }}} z ˙ = l sin θ θ ˙ {\displaystyle {\dot {z}}=l\sin \theta \,{\dot {\theta }}} . <p>Thus, </p> v 2 = x ˙ 2 + y ˙ 2 + z ˙ 2 = l 2 ( θ ˙ 2 + sin 2 θ ϕ ˙ 2 ) {\displaystyle v^{2}={\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}=l^{2}\left({\dot {\theta }}^{2}+\sin ^{2}\theta \,{\dot {\phi }}^{2}\right)} <p>and </p> T = 1 2 m v 2 = 1 2 m l 2 ( θ ˙ 2 + sin 2 θ ϕ ˙ 2 ) {\displaystyle T={\tfrac {1}{2}}mv^{2}={\tfrac {1}{2}}ml^{2}\left({\dot {\theta }}^{2}+\sin ^{2}\theta \,{\dot {\phi }}^{2}\right)} V = m g z = m g l ( 1 − cos θ ) {\displaystyle V=mg\,z=mg\,l(1-\cos \theta )} <p>The Lagrangian, with constant parts removed, is<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> L = 1 2 m l 2 ( θ ˙ 2 + sin 2 θ ϕ ˙ 2 ) + m g l cos θ . {\displaystyle L={\frac {1}{2}}ml^{2}\left({\dot {\theta }}^{2}+\sin ^{2}\theta \ {\dot {\phi }}^{2}\right)+mgl\cos \theta .} <p>The <a href="/facts/Euler%25E2%2580%2593Lagrange_equations/Ymq3r3Dh">Euler–Lagrange equation</a> involving the polar angle θ {\displaystyle \theta } </p> d d t ∂ ∂ θ ˙ L − ∂ ∂ θ L = 0 {\displaystyle {\frac {d}{dt}}{\frac {\partial }{\partial {\dot {\theta }}}}L-{\frac {\partial }{\partial \theta }}L=0} <p>gives </p> d d t ( m l 2 θ ˙ ) − m l 2 sin θ ⋅ cos θ ϕ ˙ 2 + m g l sin θ = 0 {\displaystyle {\frac {d}{dt}}\left(ml^{2}{\dot {\theta }}\right)-ml^{2}\sin \theta \cdot \cos \theta \,{\dot {\phi }}^{2}+mgl\sin \theta =0} <p>and </p> θ ¨ = sin θ cos θ ϕ ˙ 2 − g l sin θ {\displaystyle {\ddot {\theta }}=\sin \theta \cos \theta {\dot {\phi }}^{2}-{\frac {g}{l}}\sin \theta } <p>When ϕ ˙ = 0 {\displaystyle {\dot {\phi }}=0} the equation reduces to the <a href="/facts/Differential_equation/9MGbE3Ca">differential equation</a> for the motion of a <a href="/facts/Pendulum_(mathematics)/e7ldI0ot">simple gravity pendulum</a>. </p><p>Similarly, the Euler–Lagrange equation involving the azimuth ϕ {\displaystyle \phi } , </p> d d t ∂ ∂ ϕ ˙ L − ∂ ∂ ϕ L = 0 {\displaystyle {\frac {d}{dt}}{\frac {\partial }{\partial {\dot {\phi }}}}L-{\frac {\partial }{\partial \phi }}L=0} <p>gives </p> d d t ( m l 2 sin 2 θ ⋅ ϕ ˙ ) = 0 {\displaystyle {\frac {d}{dt}}\left(ml^{2}\sin ^{2}\theta \cdot {\dot {\phi }}\right)=0} . <p>The last equation shows that <a href="/facts/Angular_momentum/7TuSVFxq">angular momentum</a> around the vertical axis, | L z | = l sin θ × m l sin θ ϕ ˙ {\displaystyle |\mathbf {L} _{z}|=l\sin \theta \times ml\sin \theta \,{\dot {\phi }}} is conserved. The factor m l 2 sin 2 θ {\displaystyle ml^{2}\sin ^{2}\theta } will play a role in the Hamiltonian formulation below. </p><p>The second order differential equation determining the evolution of ϕ {\displaystyle \phi } is thus </p> ϕ ¨ sin θ = − 2 θ ˙ ϕ ˙ cos θ {\displaystyle {\ddot {\phi }}\,\sin \theta =-2\,{\dot {\theta }}\,{\dot {\phi }}\,\cos \theta } . <p>The azimuth ϕ {\displaystyle \phi } , being absent from the Lagrangian, is a <a href="/facts/Cyclic_coordinate/sTxyqWz8">cyclic coordinate</a>, which implies that its <a href="/facts/Conjugate_momentum/9g2SFFRz">conjugate momentum</a> is a <a href="/facts/Constant_of_motion/3ePkaHK6">constant of motion</a>. </p><p>The <a href="/facts/Conical_pendulum/hvdBSHTk">conical pendulum</a> refers to the special solutions where θ ˙ = 0 {\displaystyle {\dot {\theta }}=0} and ϕ ˙ {\displaystyle {\dot {\phi }}} is a constant not depending on time. </p> <h2 id="hamiltonian-mechanics">Hamiltonian mechanics</h2> <p class="note">Main article: <a href="/facts/Hamiltonian_mechanics/2ReerlNA">Hamiltonian mechanics</a></p> <p>The Hamiltonian is </p> H = P θ θ ˙ + P ϕ ϕ ˙ − L {\displaystyle H=P_{\theta }{\dot {\theta }}+P_{\phi }{\dot {\phi }}-L} <p>where conjugate momenta are </p> P θ = ∂ L ∂ θ ˙ = m l 2 ⋅ θ ˙ {\displaystyle P_{\theta }={\frac {\partial L}{\partial {\dot {\theta }}}}=ml^{2}\cdot {\dot {\theta }}} <p>and </p> P ϕ = ∂ L ∂ ϕ ˙ = m l 2 sin 2 θ ⋅ ϕ ˙ {\displaystyle P_{\phi }={\frac {\partial L}{\partial {\dot {\phi }}}}=ml^{2}\sin ^{2}\!\theta \cdot {\dot {\phi }}} . <p>In terms of coordinates and momenta it reads </p><p> H = [ 1 2 m l 2 θ ˙ 2 + 1 2 m l 2 sin 2 θ ϕ ˙ 2 ] ⏟ T + [ − m g l cos θ ] ⏟ V = P θ 2 2 m l 2 + P ϕ 2 2 m l 2 sin 2 θ − m g l cos θ {\displaystyle H=\underbrace {\left[{\frac {1}{2}}ml^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}ml^{2}\sin ^{2}\theta {\dot {\phi }}^{2}\right]} _{T}+\underbrace {{\bigg [}-mgl\cos \theta {\bigg ]}} _{V}={P_{\theta }^{2} \over 2ml^{2}}+{P_{\phi }^{2} \over 2ml^{2}\sin ^{2}\theta }-mgl\cos \theta } </p><p>Hamilton's equations will give time evolution of coordinates and momenta in four first-order differential equations </p> θ ˙ = P θ m l 2 {\displaystyle {\dot {\theta }}={P_{\theta } \over ml^{2}}} ϕ ˙ = P ϕ m l 2 sin 2 θ {\displaystyle {\dot {\phi }}={P_{\phi } \over ml^{2}\sin ^{2}\theta }} P θ ˙ = P ϕ 2 m l 2 sin 3 θ cos θ − m g l sin θ {\displaystyle {\dot {P_{\theta }}}={P_{\phi }^{2} \over ml^{2}\sin ^{3}\theta }\cos \theta -mgl\sin \theta } P ϕ ˙ = 0 {\displaystyle {\dot {P_{\phi }}}=0} <p>Momentum P ϕ {\displaystyle P_{\phi }} is a constant of motion. That is a consequence of the rotational symmetry of the system around the vertical axis.[<i>dubious – discuss</i>] </p> <h2 id="trajectory">Trajectory</h2> <p>Trajectory of the mass on the sphere can be obtained from the expression for the total energy </p> E = [ 1 2 m l 2 θ ˙ 2 + 1 2 m l 2 sin 2 θ ϕ ˙ 2 ] ⏟ T + [ − m g l cos θ ] ⏟ V {\displaystyle E=\underbrace {\left[{\frac {1}{2}}ml^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}ml^{2}\sin ^{2}\theta {\dot {\phi }}^{2}\right]} _{T}+\underbrace {{\bigg [}-mgl\cos \theta {\bigg ]}} _{V}} <p>by noting that the horizontal component of angular momentum L z = m l 2 sin 2 θ ϕ ˙ {\displaystyle L_{z}=ml^{2}\sin ^{2}\!\theta \,{\dot {\phi }}} is a constant of motion, independent of time.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> This is true because neither gravity nor the reaction from the sphere act in directions that would affect this component of angular momentum. </p><p>Hence </p> E = 1 2 m l 2 θ ˙ 2 + 1 2 L z 2 m l 2 sin 2 θ − m g l cos θ {\displaystyle E={\frac {1}{2}}ml^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}{\frac {L_{z}^{2}}{ml^{2}\sin ^{2}\theta }}-mgl\cos \theta } ( d θ d t ) 2 = 2 m l 2 [ E − 1 2 L z 2 m l 2 sin 2 θ + m g l cos θ ] {\displaystyle \left({\frac {d\theta }{dt}}\right)^{2}={\frac {2}{ml^{2}}}\left[E-{\frac {1}{2}}{\frac {L_{z}^{2}}{ml^{2}\sin ^{2}\theta }}+mgl\cos \theta \right]} <p>which leads to an <a href="/facts/Elliptic_integral/q8EGYzQ3">elliptic integral</a> of the first kind<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> for θ {\displaystyle \theta } </p> t ( θ ) = 1 2 m l 2 ∫ [ E − 1 2 L z 2 m l 2 sin 2 θ + m g l cos θ ] − 1 2 d θ {\displaystyle t(\theta )={\sqrt {{\tfrac {1}{2}}ml^{2}}}\int \left[E-{\frac {1}{2}}{\frac {L_{z}^{2}}{ml^{2}\sin ^{2}\theta }}+mgl\cos \theta \right]^{-{\frac {1}{2}}}\,d\theta } <p>and an elliptic integral of the third kind for ϕ {\displaystyle \phi } </p> ϕ ( θ ) = L z l 2 m ∫ sin − 2 θ [ E − 1 2 L z 2 m l 2 sin 2 θ + m g l cos θ ] − 1 2 d θ {\displaystyle \phi (\theta )={\frac {L_{z}}{l{\sqrt {2m}}}}\int \sin ^{-2}\theta \left[E-{\frac {1}{2}}{\frac {L_{z}^{2}}{ml^{2}\sin ^{2}\theta }}+mgl\cos \theta \right]^{-{\frac {1}{2}}}\,d\theta } . <p>The angle θ {\displaystyle \theta } lies between two circles of latitude,<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> where </p> E > 1 2 L z 2 m l 2 sin 2 θ − m g l cos θ {\displaystyle E>{\frac {1}{2}}{\frac {L_{z}^{2}}{ml^{2}\sin ^{2}\theta }}-mgl\cos \theta } . <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Foucault_pendulum/IdB9XaGt">Foucault pendulum</a></li> <li><a href="/facts/Conical_pendulum/hvdBSHTk">Conical pendulum</a></li> <li><a href="/facts/Newton%2527s_laws_of_motion/XKzGTXK1">Newton's three laws of motion</a></li> <li><a href="/facts/Pendulum/D2hQXwlI">Pendulum</a></li> <li><a href="/facts/Pendulum_(mathematics)/e7ldI0ot">Pendulum (mathematics)</a></li> <li><a href="/facts/Routhian_mechanics/ER2I9KlQ">Routhian mechanics</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Weinstein, Alexander (1942). "The spherical pendulum and complex integration". <i>The American Mathematical Monthly</i>. 49 (8): 521–523. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1080%2F00029890.1942.11991275">10.1080/00029890.1942.11991275</a>.</li> <li>Kohn, Walter (1946). <a href="https://doi.org/10.2307%2F1990314">"Contour integration in the theory of the spherical pendulum and the heavy symmetrical top"</a>. <i>Transactions of the American Mathematical Society</i>. 59 (1): 107–131. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1990314">10.2307/1990314</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1990314">1990314</a>.</li> <li>Olsson, M. G. (1981). "Spherical pendulum revisited". <i>American Journal of Physics</i>. 49 (6): 531–534. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1981AmJPh..49..531O">1981AmJPh..49..531O</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1119%2F1.12666">10.1119/1.12666</a>.</li> <li>Horozov, Emil (1993). "On the isoenergetical non-degeneracy of the spherical pendulum". <i>Physics Letters A</i>. 173 (3): 279–283. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1993PhLA..173..279H">1993PhLA..173..279H</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0375-9601%2893%2990279-9">10.1016/0375-9601(93)90279-9</a>.</li> <li>Richter, Peter H.; Dullin, Holger R.; Waalkens, Holger; Wiersig, Jan (1996). <a href="https://research.rug.nl/en/publications/8af67cd3-5599-4167-a2dd-aed857ac6bdf">"Spherical pendulum, actions and spin"</a>. <i>J. Phys. Chem</i>. 100 (49): 19124–19135. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1021%2Fjp9617128">10.1021/jp9617128</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:18023607">18023607</a>.</li> <li>Shiriaev, A. S.; Ludvigsen, H.; Egeland, O. (2004). "Swinging up the spherical pendulum via stabilization of its first integrals". <i>Automatica</i>. 40: 73–85. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.automatica.2003.07.009">10.1016/j.automatica.2003.07.009</a>.</li> <li>Essen, Hanno; Apazidis, Nicholas (2009). "Turning points of the spherical pendulum and the golden ratio". <i>European Journal of Physics</i>. 30 (2): 427–432. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2009EJPh...30..427E">2009EJPh...30..427E</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F0143-0807%2F30%2F2%2F021">10.1088/0143-0807/30/2/021</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:121216295">121216295</a>.</li> <li>Dullin, Holger R. (2013). <a href="https://doi.org/10.1016%2Fj.jde.2013.01.018">"Semi-global symplectic invariants of the spherical pendulum"</a>. <i>Journal of Differential Equations</i>. 254 (7): 2942–2963. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1108.4962">1108.4962</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2013JDE...254.2942D">2013JDE...254.2942D</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.jde.2013.01.018">10.1016/j.jde.2013.01.018</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Landau, Lev Davidovich; Evgenii Mikhailovich Lifshitz (1976). Course of Theoretical Physics: Volume 1 Mechanics. Butterworth-Heinenann. pp. 33–34. ISBN 0750628960. <a href="0750628960" target="_blank">0750628960</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Landau, Lev Davidovich; Evgenii Mikhailovich Lifshitz (1976). Course of Theoretical Physics: Volume 1 Mechanics. Butterworth-Heinenann. pp. 33–34. ISBN 0750628960. <a href="0750628960" target="_blank">0750628960</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Landau, Lev Davidovich; Evgenii Mikhailovich Lifshitz (1976). Course of Theoretical Physics: Volume 1 Mechanics. Butterworth-Heinenann. pp. 33–34. ISBN 0750628960. <a href="0750628960" target="_blank">0750628960</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Landau, Lev Davidovich; Evgenii Mikhailovich Lifshitz (1976). Course of Theoretical Physics: Volume 1 Mechanics. Butterworth-Heinenann. pp. 33–34. ISBN 0750628960. <a href="0750628960" target="_blank">0750628960</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>