Geodesic
Newton's Off-Center Circular Orbits and the Magnetic Monopole
Symmetry, integrability and geometry: methods and applications, Tome 19 (2023) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Introducing a radially dependent magnetic field into Newton's off-center circular orbits potential so as to preserve the $E=0$ dynamical symmetry leads to a unique choice of field that can be identified as the inclusion of a magnetic monopole in the inverse stereographically projected problem. One finds also a phenomenological correspondence with that of the linearly damped Kepler model. The presence of the monopole field deforms the symmetry algebra by a central extension, and the quantum mechanical version of this algebra reveals a number of zero modes equal to that counted using the index theorem of elliptic operators.
Keywords: integrals of motion, magnetic monopole
Mots-clés : zero modes. 
@article{SIGMA_2023_19_a98,
     author = {Dipesh Bhandari and Michael Crescimanno},
     title = {Newton's {Off-Center} {Circular} {Orbits} and the {Magnetic} {Monopole}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2023},
     volume = {19},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a98/}
}
TY  - JOUR
AU  - Dipesh Bhandari
AU  - Michael Crescimanno
TI  - Newton's Off-Center Circular Orbits and the Magnetic Monopole
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2023
VL  - 19
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a98/
LA  - en
ID  - SIGMA_2023_19_a98
ER  - 
%0 Journal Article
%A Dipesh Bhandari
%A Michael Crescimanno
%T Newton's Off-Center Circular Orbits and the Magnetic Monopole
%J Symmetry, integrability and geometry: methods and applications
%D 2023
%V 19
%U http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a98/
%G en
%F SIGMA_2023_19_a98
Dipesh Bhandari; Michael Crescimanno. Newton's Off-Center Circular Orbits and the Magnetic Monopole. Symmetry, integrability and geometry: methods and applications, Tome 19 (2023). http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a98/

[1] Andrade e Silva R., Jacobson T., “Particle on the sphere: group-theoretic quantization in the presence of a magnetic monopole”, J. Phys. A, 54 (2021), 235303, 33 pp., arXiv: 2011.04888 | DOI | MR | Zbl

[2] Atiyah M.F., Bott R., Patodi V.K., “On the heat equation and the index theorem”, Invent. Math., 19 (1973), 279–330 | DOI | MR | Zbl

[3] Atiyah M.F., Singer I.M., “The index of elliptic operators on compact manifolds”, Bull. Amer. Math. Soc., 69 (1963), 422–433 | DOI | MR | Zbl

[4] Bardakci K., Crescimanno M., “Monopole backgrounds on the world sheet”, Nuclear Phys. B, 313 (1989), 269–292 | DOI | MR

[5] Cooper L.N., “Bound electron pairs in a degenerate Fermi gas”, Phys. Rev., 104 (1956), 1189–1190 | DOI

[6] Dirac P., “Quantised singularities in the electromagnetic field”, Proc. Roy. Soc. Lond. A, 133 (1931), 60–72 | DOI | Zbl

[7] Faure R., “Transformations conformes en mécanique ondulatoire”, C. R. Acad. Sci. Paris, 237 (1953), 603–605 | MR | Zbl

[8] Gauss C.F., “General investigations of curved surfaces of 1827 and 1825”, Nature, 66 (1902), 316–317 | DOI | MR

[9] Golo V., “Dynamic ${\rm SO}(3,1)$ symmetry of the Dirac magnetic monopol”, JETP Lett., 35 (1982), 663–665

[10] Grossman B., “A $3$-cocyle in quantum mechanics”, Phys. Lett. B, 152 (1985), 93–97 | DOI | MR

[11] Haldane F.D.M., Rezayi E.H., “Spin-singlet wave function for the half-integral quantum Hall effect”, Phys. Rev. Lett., 60 (1988), 1886–1886 | DOI | MR

[12] Hamilton B., Crescimanno M., “Linear frictional forces cause orbits to neither circularize nor precess”, J. Phys. A, 41 (2008), 235205, 13 pp., arXiv: 0708.3827 | DOI | MR | Zbl

[13] Hamilton W.R., “The Hodograph or a new method of expressing in symbolic language the Newtonian law of attraction”, Proc. R. Ir. Acad., 3 (1847), 344–353

[14] Ince E.L., Ordinary differential equations, Dover Publications, New York, 1956 | MR

[15] Kemp G.M., Veselov A.P., “On geometric quantization of the Dirac magnetic monopole”, J. Nonlinear Math. Phys., 21 (2014), 34–42, arXiv: 1103.6242 | DOI | MR | Zbl

[16] Laughlin R.B., “Quantized Hall conductivity in two dimensions”, Phys. Rev. B, 23 (1981), 5632–5633 | DOI

[17] Lie S., Vorlesung über Differentialgleichungen mit bekannten infinitesimalen Transformationen, B.G. Teubner Verlag, 1891 | DOI

[18] Maxwell J.C., Matter and motion, Cambridge Library Collect. Phys. Sci., Cambridge University Press, 2010 | DOI

[19] McIntosh H.V., Cisneros A., “Degeneracy in the presence of a magnetic monopole”, J. Math. Phys., 11 (1970), 896–916 | DOI | MR

[20] McSween E., Winternitz P., “Integrable and superintegrable Hamiltonian systems in magnetic fields”, J. Math. Phys., 41 (2000), 2957–2967 | DOI | MR | Zbl

[21] Newton I., Newton's Principia. The mathematical principles of natural philosophy, Cambridge Library Collect. Phys. Sci., D. Adee, New-York, 1848 | MR

[22] Olshanii M., “A novel potential featuring off-center circular orbits”, SIGMA, 19 (2023), 001, 8 pp., arXiv: 2207.09606 | DOI | MR | Zbl

[23] Senthil T., Levin M., “Integer quantum Hall effect for bosons”, Phys. Rev. Lett., 110 (2013), 046801, 5 pp. | DOI

[24] Shnir Y.M., Magnetic monopoles, Texts Monogr. Phys., Springer, Berlin, 2005 | DOI | MR | Zbl

[25] Singer I.M., “Future extensions of index theory and elliptic operators”, Prospects in Mathematics, Ann. of Math. Stud., 70, Princeton University Press, Princeton, NJ, 1971, 171–185 | DOI | MR

[26] Song H., Jo S.G., “Quantum mechanics on $S^1$, $S^2$ and Lorentz group”, J. Korean Phys. Soc., 59 (2011), 3314–3320 | DOI

[27] Suzuki M.S., Suzuki I.S., Laplace–Runge–Lenz triangles in Feynman hodograph diagram: the Kepler's model and Sommerfeld's model, Binghamton, New York, 2022