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Hamiltonian mechanics
Main article: Hamiltonian mechanics
The Hamiltonian is

{\displaystyle H=P_{\theta }{\dot {\theta }}+P_{\phi }{\dot {\phi }}-L}H=P_{\theta }{\dot {\theta }}+P_{\phi }{\dot {\phi }}-L
where conjugate momenta are

{\displaystyle P_{\theta }={\frac {\partial L}{\partial {\dot {\theta }}}}=ml^{2}\cdot {\dot {\theta }}}{\displaystyle P_{\theta }={\frac {\partial L}{\partial {\dot {\theta }}}}=ml^{2}\cdot {\dot {\theta }}}
and

{\displaystyle P_{\phi }={\frac {\partial L}{\partial {\dot {\phi }}}}=ml^{2}\sin ^{2}\!\theta \cdot {\dot {\phi }}}{\displaystyle P_{\phi }={\frac {\partial L}{\partial {\dot {\phi }}}}=ml^{2}\sin ^{2}\!\theta \cdot {\dot {\phi }}}.
In terms of coordinates and momenta it reads

{\displaystyle H=\underbrace {{\Big [}{\frac {1}{2}}ml^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}ml^{2}\sin ^{2}\theta {\dot {\phi }}^{2}{\Big ]}} _{T}+\underbrace {{\Big [}-mgl\cos \theta {\Big ]}} _{V}={P_{\theta }^{2} \over 2ml^{2}}+{P_{\phi }^{2} \over 2ml^{2}\sin ^{2}\theta }-mgl\cos \theta }{\displaystyle H=\underbrace {{\Big [}{\frac {1}{2}}ml^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}ml^{2}\sin ^{2}\theta {\dot {\phi }}^{2}{\Big ]}} _{T}+\underbrace {{\Big [}-mgl\cos \theta {\Big ]}} _{V}={P_{\theta }^{2} \over 2ml^{2}}+{P_{\phi }^{2} \over 2ml^{2}\sin ^{2}\theta }-mgl\cos \theta }

Hamilton's equations will give time evolution of coordinates and momenta in four first-order differential equations

{\displaystyle {\dot {\theta }}={P_{\theta } \over ml^{2}}}{\displaystyle {\dot {\theta }}={P_{\theta } \over ml^{2}}}
{\displaystyle {\dot {\phi }}={P_{\phi } \over ml^{2}\sin ^{2}\theta }}{\displaystyle {\dot {\phi }}={P_{\phi } \over ml^{2}\sin ^{2}\theta }}
{\displaystyle {\dot {P_{\theta }}}={P_{\phi }^{2} \over ml^{2}\sin ^{3}\theta }\cos \theta -mgl\sin \theta }{\displaystyle {\dot {P_{\theta }}}={P_{\phi }^{2} \over ml^{2}\sin ^{3}\theta }\cos \theta -mgl\sin \theta }
{\displaystyle {\dot {P_{\phi }}}=0}{\displaystyle {\dot {P_{\phi }}}=0}
Momentum {\displaystyle P_{\phi }}{\displaystyle P_{\phi }} is a constant of motion. That is a consequence of the rotational symmetry of the system around the vertical axis.[dubious – discuss]

Trajectory

Trajectory of a spherical pendulum.
Trajectory of the mass on the sphere can be obtained from the expression for the total energy

{\displaystyle E=\underbrace {{\Big [}{\frac {1}{2}}ml^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}ml^{2}\sin ^{2}\theta {\dot {\phi }}^{2}{\Big ]}} _{T}+\underbrace {{\Big [}-mgl\cos \theta {\Big ]}} _{V}}{\displaystyle E=\underbrace {{\Big [}{\frac {1}{2}}ml^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}ml^{2}\sin ^{2}\theta {\dot {\phi }}^{2}{\Big ]}} _{T}+\underbrace {{\Big [}-mgl\cos \theta {\Big ]}} _{V}}
by noting that the vertical component of angular momentum {\displaystyle L_{z}=ml^{2}\sin ^{2}\!\theta \,{\dot {\phi }}}{\displaystyle L_{z}=ml^{2}\sin ^{2}\!\theta \,{\dot {\phi }}} is a constant of motion, independent of time.[1]

Hence

{\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 }{\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 }
{\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]}{\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]}
which leads to an elliptic integral of the first kind[1] for {\displaystyle \theta }\theta

{\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 }{\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 }
and an elliptic integral of the third kind for {\displaystyle \phi }\phi

{\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 }{\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 }.
The angle {\displaystyle \theta }\theta lies between two circles of latitude,[1] where

{\displaystyle E>{\frac {1}{2}}{\frac {L_{z}^{2}}{ml^{2}\sin ^{2}\theta }}-mgl\cos \theta }{\displaystyle E>{\frac {1}{2}}{\frac {L_{z}^{2}}{ml^{2}\sin ^{2}\theta }}-mgl\cos \theta }.
See also
Foucault pendulum
Conical pendulum
Newton's three laws of motion
Pendulum
Pendulum (mathematics)
Routhian mechanics
References
Landau, Lev Davidovich; Evgenii Mikhailovich Lifshitz (1976). Course of Theoretical Physics: Volume 1 Mechanics. Butterworth-Heinenann. pp. 33–34. ISBN 0750628960.
Further reading
Weinstein, Alexander (1942). "The spherical pendulum and complex integration". The American Mathematical Monthly. 49 (8): 521–523. doi:10.1080/00029890.1942.11991275.
Kohn, Walter (1946). "Countour integration in the theory of the spherical pendulum and the heavy symmetrical top". Transactions of the American Mathematical Society. 59 (1): 107–131. doi:10.2307/1990314. JSTOR 1990314.
Olsson, M. G. (1981). "Spherical pendulum revisited". American Journal of Physics. 49 (6): 531–534. Bibcode:1981AmJPh..49..531O. doi:10.1119/1.12666.
Horozov, Emil (1993). "On the isoenergetical non-degeneracy of the spherical pendulum". Physics Letters A. 173 (3): 279–283. Bibcode:1993PhLA..173..279H. doi:10.1016/0375-9601(93)90279-9.
Shiriaev, A. S.; Ludvigsen, H.; Egeland, O. (2004). "Swinging up the spherical pendulum via stabilizatio of its first integrals". Automatica. 40: 73–85. doi:10.1016/j.automatica.2003.07.009.
Essen, Hanno; Apazidis, Nicholas (2009). "Turning points of the spherical pendulum and the golden ration". European Journal of Physics. 30 (2): 427–432. Bibcode:2009EJPh...30..427E. doi:10.1088/0143-0807/30/2/021.
Dullin, Holger R. (2013). "Semi-global symplectic invariants of the spherical pendulum". Journal of Differential Equations. 254 (7): 2942–2963. doi:10.1016/j.jde.2013.01.018.
Categories: Pendulums
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This page was last edited on 17 December 2021, at 04:56 (UTC).
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halilo	#18075 https://en.wikipedia.org/wiki/Spherical_pendulum#Hamiltonian_mechanics is imposible to read :\|
halilo	#18076 Hamiltonian mechanics Main article: Hamiltonian mechanics The Hamiltonian is {\displaystyle H=P_{\theta }{\dot {\theta }}+P_{\phi }{\dot {\phi }}-L}H=P_{\theta }{\dot {\theta }}+P_{\phi }{\dot {\phi }}-L where conjugate momenta are {\displaystyle P_{\theta }={\frac {\partial L}{\partial {\dot {\theta }}}}=ml^{2}\cdot {\dot {\theta }}}{\displaystyle P_{\theta }={\frac {\partial L}{\partial {\dot {\theta }}}}=ml^{2}\cdot {\dot {\theta }}} and {\displaystyle P_{\phi }={\frac {\partial L}{\partial {\dot {\phi }}}}=ml^{2}\sin ^{2}\!\theta \cdot {\dot {\phi }}}{\displaystyle P_{\phi }={\frac {\partial L}{\partial {\dot {\phi }}}}=ml^{2}\sin ^{2}\!\theta \cdot {\dot {\phi }}}. In terms of coordinates and momenta it reads {\displaystyle H=\underbrace {{\Big [}{\frac {1}{2}}ml^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}ml^{2}\sin ^{2}\theta {\dot {\phi }}^{2}{\Big ]}} _{T}+\underbrace {{\Big [}-mgl\cos \theta {\Big ]}} _{V}={P_{\theta }^{2} \over 2ml^{2}}+{P_{\phi }^{2} \over 2ml^{2}\sin ^{2}\theta }-mgl\cos \theta }{\displaystyle H=\underbrace {{\Big [}{\frac {1}{2}}ml^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}ml^{2}\sin ^{2}\theta {\dot {\phi }}^{2}{\Big ]}} _{T}+\underbrace {{\Big [}-mgl\cos \theta {\Big ]}} _{V}={P_{\theta }^{2} \over 2ml^{2}}+{P_{\phi }^{2} \over 2ml^{2}\sin ^{2}\theta }-mgl\cos \theta } Hamilton's equations will give time evolution of coordinates and momenta in four first-order differential equations {\displaystyle {\dot {\theta }}={P_{\theta } \over ml^{2}}}{\displaystyle {\dot {\theta }}={P_{\theta } \over ml^{2}}} {\displaystyle {\dot {\phi }}={P_{\phi } \over ml^{2}\sin ^{2}\theta }}{\displaystyle {\dot {\phi }}={P_{\phi } \over ml^{2}\sin ^{2}\theta }} {\displaystyle {\dot {P_{\theta }}}={P_{\phi }^{2} \over ml^{2}\sin ^{3}\theta }\cos \theta -mgl\sin \theta }{\displaystyle {\dot {P_{\theta }}}={P_{\phi }^{2} \over ml^{2}\sin ^{3}\theta }\cos \theta -mgl\sin \theta } {\displaystyle {\dot {P_{\phi }}}=0}{\displaystyle {\dot {P_{\phi }}}=0} Momentum {\displaystyle P_{\phi }}{\displaystyle P_{\phi }} is a constant of motion. That is a consequence of the rotational symmetry of the system around the vertical axis.[dubious – discuss] Trajectory Trajectory of a spherical pendulum. Trajectory of the mass on the sphere can be obtained from the expression for the total energy {\displaystyle E=\underbrace {{\Big [}{\frac {1}{2}}ml^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}ml^{2}\sin ^{2}\theta {\dot {\phi }}^{2}{\Big ]}} _{T}+\underbrace {{\Big [}-mgl\cos \theta {\Big ]}} _{V}}{\displaystyle E=\underbrace {{\Big [}{\frac {1}{2}}ml^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}ml^{2}\sin ^{2}\theta {\dot {\phi }}^{2}{\Big ]}} _{T}+\underbrace {{\Big [}-mgl\cos \theta {\Big ]}} _{V}} by noting that the vertical component of angular momentum {\displaystyle L_{z}=ml^{2}\sin ^{2}\!\theta \,{\dot {\phi }}}{\displaystyle L_{z}=ml^{2}\sin ^{2}\!\theta \,{\dot {\phi }}} is a constant of motion, independent of time.[1] Hence {\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 }{\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 } {\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]}{\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]} which leads to an elliptic integral of the first kind[1] for {\displaystyle \theta }\theta {\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 }{\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 } and an elliptic integral of the third kind for {\displaystyle \phi }\phi {\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 }{\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 }. The angle {\displaystyle \theta }\theta lies between two circles of latitude,[1] where {\displaystyle E>{\frac {1}{2}}{\frac {L_{z}^{2}}{ml^{2}\sin ^{2}\theta }}-mgl\cos \theta }{\displaystyle E>{\frac {1}{2}}{\frac {L_{z}^{2}}{ml^{2}\sin ^{2}\theta }}-mgl\cos \theta }. See also Foucault pendulum Conical pendulum Newton's three laws of motion Pendulum Pendulum (mathematics) Routhian mechanics References Landau, Lev Davidovich; Evgenii Mikhailovich Lifshitz (1976). Course of Theoretical Physics: Volume 1 Mechanics. Butterworth-Heinenann. pp. 33–34. ISBN 0750628960. Further reading Weinstein, Alexander (1942). "The spherical pendulum and complex integration". The American Mathematical Monthly. 49 (8): 521–523. doi:10.1080/00029890.1942.11991275. Kohn, Walter (1946). "Countour integration in the theory of the spherical pendulum and the heavy symmetrical top". Transactions of the American Mathematical Society. 59 (1): 107–131. doi:10.2307/1990314. JSTOR 1990314. Olsson, M. G. (1981). "Spherical pendulum revisited". American Journal of Physics. 49 (6): 531–534. Bibcode:1981AmJPh..49..531O. doi:10.1119/1.12666. Horozov, Emil (1993). "On the isoenergetical non-degeneracy of the spherical pendulum". Physics Letters A. 173 (3): 279–283. Bibcode:1993PhLA..173..279H. doi:10.1016/0375-9601(93)90279-9. Shiriaev, A. S.; Ludvigsen, H.; Egeland, O. (2004). "Swinging up the spherical pendulum via stabilizatio of its first integrals". Automatica. 40: 73–85. doi:10.1016/j.automatica.2003.07.009. Essen, Hanno; Apazidis, Nicholas (2009). "Turning points of the spherical pendulum and the golden ration". European Journal of Physics. 30 (2): 427–432. Bibcode:2009EJPh...30..427E. doi:10.1088/0143-0807/30/2/021. Dullin, Holger R. (2013). "Semi-global symplectic invariants of the spherical pendulum". Journal of Differential Equations. 254 (7): 2942–2963. doi:10.1016/j.jde.2013.01.018. Categories: Pendulums Navigation menu Not logged in Talk Contributions Create account Log in ArticleTalk ReadEditView history Search Search Wikipedia Main page Contents Current events Random article About Wikipedia Contact us Donate Contribute Help Learn to edit Community portal Recent changes Upload file Tools What links here Related changes Special pages Permanent link Page information Cite this page Wikidata item Print/export Download as PDF Printable version Languages Deutsch Esperanto Français Polski Shqip Edit links This page was last edited on 17 December 2021, at 04:56 (UTC). Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
halilo	#18077 thats https://en.wikipedia.org/wiki/Spherical_pendulum#Hamiltonian_mechanics
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