Superintegrability and time-dependent integrals

Ondřej Kubů; Libor Šnobl

Archivum Mathematicum (2019)

  • Volume: 055, Issue: 5, page 309-318
  • ISSN: 0044-8753

Abstract

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While looking for additional integrals of motion of several minimally superintegrable systems in static electric and magnetic fields, we have realized that in some cases Lie point symmetries of Euler-Lagrange equations imply existence of explicitly time-dependent integrals of motion through Noether’s theorem. These integrals can be combined to get an additional time-independent integral for some values of the parameters of the considered systems, thus implying maximal superintegrability. Even for values of the parameters for which the systems don’t exhibit maximal superintegrability in the usual sense they allow a completely algebraic determination of the trajectories (including their time dependence).

How to cite

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Kubů, Ondřej, and Šnobl, Libor. "Superintegrability and time-dependent integrals." Archivum Mathematicum 055.5 (2019): 309-318. <http://eudml.org/doc/294219>.

@article{Kubů2019,
abstract = {While looking for additional integrals of motion of several minimally superintegrable systems in static electric and magnetic fields, we have realized that in some cases Lie point symmetries of Euler-Lagrange equations imply existence of explicitly time-dependent integrals of motion through Noether’s theorem. These integrals can be combined to get an additional time-independent integral for some values of the parameters of the considered systems, thus implying maximal superintegrability. Even for values of the parameters for which the systems don’t exhibit maximal superintegrability in the usual sense they allow a completely algebraic determination of the trajectories (including their time dependence).},
author = {Kubů, Ondřej, Šnobl, Libor},
journal = {Archivum Mathematicum},
keywords = {integrability; superintegrability; classical mechanics; magnetic field; time-dependent integrals},
language = {eng},
number = {5},
pages = {309-318},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Superintegrability and time-dependent integrals},
url = {http://eudml.org/doc/294219},
volume = {055},
year = {2019},
}

TY - JOUR
AU - Kubů, Ondřej
AU - Šnobl, Libor
TI - Superintegrability and time-dependent integrals
JO - Archivum Mathematicum
PY - 2019
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 055
IS - 5
SP - 309
EP - 318
AB - While looking for additional integrals of motion of several minimally superintegrable systems in static electric and magnetic fields, we have realized that in some cases Lie point symmetries of Euler-Lagrange equations imply existence of explicitly time-dependent integrals of motion through Noether’s theorem. These integrals can be combined to get an additional time-independent integral for some values of the parameters of the considered systems, thus implying maximal superintegrability. Even for values of the parameters for which the systems don’t exhibit maximal superintegrability in the usual sense they allow a completely algebraic determination of the trajectories (including their time dependence).
LA - eng
KW - integrability; superintegrability; classical mechanics; magnetic field; time-dependent integrals
UR - http://eudml.org/doc/294219
ER -

References

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  1. Crampin, M., 10.1007/BF01807231, Internat. J. Theoret. Phys. 16 (10) (1977), 741–754. (1977) MR0502454DOI10.1007/BF01807231
  2. Gubbiotti, G., Nucci, M.C., 10.1063/1.4974264, J. Math. Phys. 58 (1) (2017), 14 pp., 012902. (2017) MR3600036DOI10.1063/1.4974264
  3. Jovanović, B., 10.3934/jgm.2018006, J. Geom. Mech. 10 (2) (2018), 173–187. (2018) MR3808246DOI10.3934/jgm.2018006
  4. Lie, S., Theorie der Transformationsgruppen, Teil I–III, Leipzig: Teubner, 1888, 1890, 1893. (1888) 
  5. López, C., Martínez, E., Rañada, M.F., 10.1088/0305-4470/32/7/013, J. Phys. A 32 (7) (1999), 1241–1249. (1999) MR1690665DOI10.1088/0305-4470/32/7/013
  6. Marchesiello, A., Šnobl, L., 10.1088/1751-8121/aa6f68, J. Phys. A 50 (24) (2017), 24 pp., 245202. (2017) MR3659131DOI10.1088/1751-8121/aa6f68
  7. Marchesiello, A., Šnobl, L., An infinite family of maximally superintegrable systems in a magnetic field with higher order integrals, SIGMA Symmetry Integrability Geom. Methods Appl. 14 (092) (2018), 11 pp. (2018) MR3849972
  8. Mariwalla, K.H., 10.1088/0305-4470/13/9/002, J. Phys. A 13 (9) (1980), 289–293. (1980) MR0586473DOI10.1088/0305-4470/13/9/002
  9. Nucci, M.C., Leach, P.G.L., 10.1063/1.1337614, J. Math. Phys. 42 (2) (2001), 746–764. (2001) MR1809250DOI10.1063/1.1337614
  10. Olver, P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1993, second edition. (1993) MR1240056
  11. Prince, G., Toward a classification of dynamical symmetries in Lagrangian systems, Proceedings of the IUTAM-ISIMM symposium on modern developments in analytical mechanics, Vol. II (Torino, 1982, vol. 117, 1983, pp. 687–691. (1983) MR0773518
  12. Sarlet, W., Cantrijn, F., 10.1137/1023098, SIAM Rev. 23 (4) (1981), 467–494. (1981) Zbl0474.70014MR0636081DOI10.1137/1023098
  13. Sarlet, W., Cantrijn, F., 10.1088/0305-4470/14/2/023, J. Phys. A 14 (2) (1981), 479–492. (1981) Zbl0464.58010MR0601885DOI10.1088/0305-4470/14/2/023

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