Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
integrability; superintegrability; classical mechanics; magnetic field; time-dependent integrals
integrability; superintegrability; classical mechanics; magnetic field; time-dependent integrals
Summary:
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).
References:
[1] Crampin, M.: Constants of the motion in Lagrangian mechanics. Internat. J. Theoret. Phys. 16 (10) (1977), 741–754. DOI 10.1007/BF01807231 | MR 0502454
[2] Gubbiotti, G., Nucci, M.C.: Are all classical superintegrable systems in two-dimensional space linearizable?. J. Math. Phys. 58 (1) (2017), 14 pp., 012902. DOI 10.1063/1.4974264 | MR 3600036
[3] Jovanović, B.: Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems. J. Geom. Mech. 10 (2) (2018), 173–187. DOI 10.3934/jgm.2018006 | MR 3808246
[4] Lie, S.: Theorie der Transformationsgruppen, Teil I–III. Leipzig: Teubner, 1888, 1890, 1893.
[5] López, C., Martínez, E., Rañada, M.F.: Dynamical symmetries, non-Cartan symmetries and superintegrability of the $n$-dimensional harmonic oscillator. J. Phys. A 32 (7) (1999), 1241–1249. DOI 10.1088/0305-4470/32/7/013 | MR 1690665
[6] Marchesiello, A., Šnobl, L.: Superintegrable 3D systems in a magnetic field corresponding to Cartesian separation of variables. J. Phys. A 50 (24) (2017), 24 pp., 245202. DOI 10.1088/1751-8121/aa6f68 | MR 3659131
[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. MR 3849972
[8] Mariwalla, K.H.: A complete set of integrals in nonrelativistic mechanics. J. Phys. A 13 (9) (1980), 289–293. DOI 10.1088/0305-4470/13/9/002 | MR 0586473
[9] Nucci, M.C., Leach, P.G.L.: The harmony in the Kepler and related problems. J. Math. Phys. 42 (2) (2001), 746–764. DOI 10.1063/1.1337614 | MR 1809250
[10] Olver, P.J.: Applications of Lie groups to differential equations. Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1993, second edition. MR 1240056
[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. MR 0773518
[12] Sarlet, W., Cantrijn, F.: Generalizations of Noether’s theorem in classical mechanics. SIAM Rev. 23 (4) (1981), 467–494. DOI 10.1137/1023098 | MR 0636081 | Zbl 0474.70014
[13] Sarlet, W., Cantrijn, F.: Higher-order Noether symmetries and constants of the motion. J. Phys. A 14 (2) (1981), 479–492. DOI 10.1088/0305-4470/14/2/023 | MR 0601885 | Zbl 0464.58010
Search
Partner of
