# zbMATH — the first resource for mathematics

Superintegrability and time-dependent integrals. (English) Zbl 07144745
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).
##### MSC:
 37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
Full Text:
##### References:
 [1] Crampin, M., Constants of the motion in Lagrangian mechanics, Internat. J. Theoret. Phys. 16 (10) (1977), 741-754 [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 [3] Jovanović, B., Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems, J. Geom. Mech. 10 (2) (2018), 173-187 [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 [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 [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 [8] Mariwalla, K. H., A complete set of integrals in nonrelativistic mechanics, J. Phys. A 13 (9) (1980), 289-293 [9] Nucci, M. C.; Leach, P. G.L., The harmony in the Kepler and related problems, J. Math. Phys. 42 (2) (2001), 746-764 [10] Olver, P. J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1993, second edition [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 [12] Sarlet, W.; Cantrijn, F., Generalizations of Noether’s theorem in classical mechanics, SIAM Rev. 23 (4) (1981), 467-494 [13] Sarlet, W.; Cantrijn, F., Higher-order Noether symmetries and constants of the motion, J. Phys. A 14 (2) (1981), 479-492
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.