Figure 1.
The graph of $H\left( {x,y} \right) = \frac{1}{2}{y^2} - \cos x + 1$ with $\left( {x,y} \right) \in \left[ { - \mathit{\boldsymbol{\pi }},\mathit{\boldsymbol{\pi }}} \right] \times {\mathbb{R}}$
-
Previous Article
On some aspects of the geometry of non integrable distributions and applications
- JGM Home
- This Issue
-
Next Article
On motions without falling of an inverted pendulum with dry friction
December
2018, 10(4): 419-443.
doi: 10.3934/jgm.2018016
Bohr-Sommerfeld-Heisenberg quantization of the mathematical pendulum
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4, Canada |
In this paper we give the Bohr-Sommerfeld-Heisenberg quantization of the mathematical pendulum.
Keywords:
Bohr-Sommerfeld-Heisenberg quantization,
mathematical pendulum,
shifting operator,
geometric quantization.
Mathematics Subject Classification:
Primary: 58F15, 58F17; Secondary: 53C35.
Citation:
Richard Cushman, Jędrzej Śniatycki. Bohr-Sommerfeld-Heisenberg quantization of the mathematical pendulum. Journal of Geometric Mechanics,
2018, 10
(4)
: 419-443.
doi: 10.3934/jgm.2018016
![]()
References:
| [1] |
N. Bohr, On the constitution of atoms and molecules (part Ⅰ), Philosophical Magazine, 26 (1913), 1-25. Google Scholar |
| [2] |
R. H. Cushman and L. M. Bates,
Global Aspects of Classical Integrable Systems, second edition, Birkhäuser, Basel, 2015.
doi: 10.1007/978-3-0348-0918-4. |
| [3] |
R. Cushman and J. Śniatycki,
Bohr-Sommerfeld-Heisenberg theory in geometric quantization, J. Fixed Point Theory Appl., 13 (2013), 3-24.
doi: 10.1007/s11784-013-0118-3. |
| [4] |
R. Cushman and J. Śniatycki, Bohr-Sommerfeld Heisenberg quantization of the $2$-dimensional harmonic oscillator, arXiv: 1207.1477. Google Scholar |
| [5] |
R. Cushman and J. Śniatycki, Shifting operators in geometric quantization, arXiv: 1808.04002. Google Scholar |
| [6] |
P. A. M. Dirac, The fundamental equations of quantum mechanics, Proc. Roy. Soc. London, 109 (1925), 642-653. Google Scholar |
| [7] |
P. A. M. Dirac,
The Principles of Quantum Mechanics, 3d ed. Oxford, at the Clarendon Press, 1947. |
| [8] |
H. Dullin,
Semi-global symplectic invariants of the spherical pendulum, J. Differential Equations, 254 (2013), 2942-2963.
doi: 10.1016/j.jde.2013.01.018. |
| [9] |
W. Heisenberg, Über die quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, (German) [On the quantum theoretical meaning of kinematic and mechanical relationship], Z. Phys., 33 (1925), 879-893. Google Scholar |
| [10] |
J. Śniatycki,
Geometric Quantization and Quantum Mechanics, Applied Mathematical Series 30 Springer Verlag, New York, 1980. |
| [11] |
A. Sommerfeld, Zur Theorie der Balmerschen Serie, (German) [On the theory of the Balmer series], Sitzungberichte der Bayerischen Akademie der Wissenschaften (Mü nchen), mathematisch-physikalische Klasse, (1915), 425-458. Google Scholar |
show all references
References:
| [1] |
N. Bohr, On the constitution of atoms and molecules (part Ⅰ), Philosophical Magazine, 26 (1913), 1-25. Google Scholar |
| [2] |
R. H. Cushman and L. M. Bates,
Global Aspects of Classical Integrable Systems, second edition, Birkhäuser, Basel, 2015.
doi: 10.1007/978-3-0348-0918-4. |
| [3] |
R. Cushman and J. Śniatycki,
Bohr-Sommerfeld-Heisenberg theory in geometric quantization, J. Fixed Point Theory Appl., 13 (2013), 3-24.
doi: 10.1007/s11784-013-0118-3. |
| [4] |
R. Cushman and J. Śniatycki, Bohr-Sommerfeld Heisenberg quantization of the $2$-dimensional harmonic oscillator, arXiv: 1207.1477. Google Scholar |
| [5] |
R. Cushman and J. Śniatycki, Shifting operators in geometric quantization, arXiv: 1808.04002. Google Scholar |
| [6] |
P. A. M. Dirac, The fundamental equations of quantum mechanics, Proc. Roy. Soc. London, 109 (1925), 642-653. Google Scholar |
| [7] |
P. A. M. Dirac,
The Principles of Quantum Mechanics, 3d ed. Oxford, at the Clarendon Press, 1947. |
| [8] |
H. Dullin,
Semi-global symplectic invariants of the spherical pendulum, J. Differential Equations, 254 (2013), 2942-2963.
doi: 10.1016/j.jde.2013.01.018. |
| [9] |
W. Heisenberg, Über die quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, (German) [On the quantum theoretical meaning of kinematic and mechanical relationship], Z. Phys., 33 (1925), 879-893. Google Scholar |
| [10] |
J. Śniatycki,
Geometric Quantization and Quantum Mechanics, Applied Mathematical Series 30 Springer Verlag, New York, 1980. |
| [11] |
A. Sommerfeld, Zur Theorie der Balmerschen Serie, (German) [On the theory of the Balmer series], Sitzungberichte der Bayerischen Akademie der Wissenschaften (Mü nchen), mathematisch-physikalische Klasse, (1915), 425-458. Google Scholar |
| [1] |
Gong Chen, Peter J. Olver. Numerical simulation of nonlinear dispersive quantization. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 991-1008. doi: 10.3934/dcds.2014.34.991 |
| [2] |
Dmitry Tamarkin. Quantization of Poisson structures on R^2. Electronic Research Announcements, 1997, 3: 119-120. |
| [3] |
Mrinal Kanti Roychowdhury. Quantization coefficients for ergodic measures on infinite symbolic space. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2829-2846. doi: 10.3934/dcds.2014.34.2829 |
| [4] |
Setsuro Fujiié, Jens Wittsten. Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3851-3873. doi: 10.3934/dcds.2018167 |
| [5] |
Mrinal Kanti Roychowdhury. Least upper bound of the exact formula for optimal quantization of some uniform Cantor distributions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4555-4570. doi: 10.3934/dcds.2018199 |
| [6] |
Mikaela Iacobelli. Asymptotic analysis for a very fast diffusion equation arising from the 1D quantization problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 4929-4943. doi: 10.3934/dcds.2019201 |
| [7] |
Dmitrii Rachinskii. On geometric conditions for reduction of the Moreau sweeping process to the Prandtl-Ishlinskii operator. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3361-3386. doi: 10.3934/dcdsb.2018246 |
| [8] |
Joachim Escher, Tony Lyons. Two-component higher order Camassa-Holm systems with fractional inertia operator: A geometric approach. Journal of Geometric Mechanics, 2015, 7 (3) : 281-293. doi: 10.3934/jgm.2015.7.281 |
| [9] |
Stefano Marò. Relativistic pendulum and invariant curves. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1139-1162. doi: 10.3934/dcds.2015.35.1139 |
| [10] |
Heping Liu, Yu Liu. Refinable functions on the Heisenberg group. Communications on Pure & Applied Analysis, 2007, 6 (3) : 775-787. doi: 10.3934/cpaa.2007.6.775 |
| [11] |
Chunpeng Wang, Jingxue Yin, Bibo Lu. Anti-shifting phenomenon of a convective nonlinear diffusion equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1211-1236. doi: 10.3934/dcdsb.2010.14.1211 |
| [12] |
Chengxia Lei, Yihong Du. Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 895-911. doi: 10.3934/dcdsb.2017045 |
| [13] |
Yueding Yuan, Yang Wang, Xingfu Zou. Spatial dynamics of a Lotka-Volterra model with a shifting habitat. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5633-5671. doi: 10.3934/dcdsb.2019076 |
| [14] |
Xianwei Chen, Zhujun Jing, Xiangling Fu. Chaos control in a pendulum system with excitations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 373-383. doi: 10.3934/dcdsb.2015.20.373 |
| [15] |
Mari Paz Calvo, Jesus M. Sanz-Serna. Carrying an inverted pendulum on a bumpy road. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 429-438. doi: 10.3934/dcdsb.2010.14.429 |
| [16] |
Jean-Francois Bertazzon. Symbolic approach and induction in the Heisenberg group. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1209-1229. doi: 10.3934/dcds.2012.32.1209 |
| [17] |
Jean-Marie Souriau. On Geometric Mechanics. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 595-607. doi: 10.3934/dcds.2007.19.595 |
| [18] |
Antonio Pumariño, Claudia Valls. On the double pendulum: An example of double resonant situations. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 413-448. doi: 10.3934/dcds.2004.11.413 |
| [19] |
Ivan Polekhin. On motions without falling of an inverted pendulum with dry friction. Journal of Geometric Mechanics, 2018, 10 (4) : 411-417. doi: 10.3934/jgm.2018015 |
| [20] |
Roman Srzednicki. On periodic solutions in the Whitney's inverted pendulum problem. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2127-2141. doi: 10.3934/dcdss.2019137 |
2018 Impact Factor: 0.525
Tools
Metrics
Other articles
by authors
[Back to Top]