Figure 1.
The graph of $H\left( {x,y} \right) = \frac{1}{2}{y^2} - \cos x + 1$ with $\left( {x,y} \right) \in \left[ { - \mathit{\boldsymbol{\pi }},\mathit{\boldsymbol{\pi }}} \right] \times {\mathbb{R}}$
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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
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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 |
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