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Russia Moscow
Section Mechanics
Title Influence of Bartnett-London and Einstein-de Haas effects on the motion of the nonholonomic sphere of Routh
Author(-s) Borisov A.V.a, Tsiganov A.V.a
Affiliations Steklov Mathematical Institute, Russian Academy of Sciencesa
Abstract We consider the rolling of an unbalanced dynamically symmetric ball along a plane without slipping in the presence of an external magnetic field. We assume that the ball may be wholly or partially composed of dielectric, ferromagnetic, or superconducting materials. According to the existing phenomenological theory, in this case, when studying the dynamics of a ball, it is required to take into account the Lorentz force moment, Barnett-London moment, and Einstein-de Haas moment. Within the framework of this mathematical model, we obtain the conditions for the existence of integrals of motion, which allow us to reduce the integration of equations of motion to a quadrature similar to the Lagrange quadrature for a heavy rigid body.
Keywords nonholonomic systems, magnetic field, integrable systems
UDC 531.011, 537.634
MSC 37J60, 70F25, 74F15
DOI 10.20537/vm190409
Received 1 November 2019
Language Russian
Citation Borisov A.V., Tsiganov A.V. Influence of Bartnett-London and Einstein-de Haas effects on the motion of the nonholonomic sphere of Routh, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 4, pp. 583-598.
  1. Ampere A.M. Sur deux Mémoires lus par M. Ampère à l'Académie royale des Sciences, Journal de Physique, de Chimie et d'Histoire Naturelle, 1821, vol. 92, pp. 160-165.
  2. Barnett S.J. On magnetization by angular acceleration, Science, 1909, vol. 30, issue 769, pp. 413.
  3. Barnett S.J. Magnetization by rotation, Physical Review, 1915, vol. 6, issue 4, pp. 239-270.
  4. Barnett S.J. Gyromagnetic and electron-inertia effects, Reviews of Modern Physics, 1935, vol. 7, issue 2, pp. 129-166.
  5. Bai Y., Svinin M., Yamamoto M. Dynamics-based motion planning for a pendulum-actuated spherical rolling robot, Regular and Chaotic Dynamics, 2018, vol. 23, issue 4, pp. 372-388.
  6. Becker R., Heller G., Sauter F. Über die Stromverteilung in einer supraleitenden Kugel, Zeitschrift für Physik, 1933, vol. 85, issue 11-12, pp. 772-787.
  7. Bizayev I.A., Tsiganov A.V. On the Routh sphere problem, Journal of Physics A: Mathematical and Theoretical, 2013, vol. 46, issue 8, 085202.
  8. Bizyaev I.A. The inertial motion of a roller racer, Regular and Chaotic Dynamics, 2017, vol. 22, issue 3, pp. 239-247.
  9. Bizyaev I.A., Borisov A.V., Mamaev I.S. The Chaplygin sleigh with parametric excitation: chaotic dynamics and nonholonomic acceleration, Regular and Chaotic Dynamics, 2017, vol. 22, issue 8, pp. 955-975.
  10. Bizyaev I.A., Borisov A.V., Mamaev I.S. An invariant measure and the probability of a fall in the problem of an inhomogeneous disk rolling on a plane, Regular and Chaotic Dynamics, 2018, vol. 23, issue 6, pp. 665-684.
  11. Bizyaev I.A., Borisov A.V., Mamaev I.S. Exotic dynamics of nonholonomic roller racer with periodic control, Regular and Chaotic Dynamics, 2018, vol. 23, issue 7-8, pp. 983-994.
  12. Borisov A.V., Mamaev I.S., Tsiganov A.V. Non-holonomic dynamics and Poisson geometry, Russian Mathematical Surveys, 2014, vol. 69, no. 3, pp. 481-538.
  13. Borisov A.V., Mamaev I.S. Rigid body dynamics, De Gruyter Stud. Math. Phys., vol. 52, Berlin: De Gruyter, 2018.
  14. Borisov A.V., Kuznetsov S.P. Comparing dynamics initiated by an attached oscillating particle for the nonholonomic model of a Chaplygin sleigh and for a model with strong transverse and weak longitudinal viscous friction applied at a fixed point on the body, Regular and Chaotic Dynamics, 2018, vol. 23, issue 7-8, pp. 803-820.
  15. Borisov A.V., Tsiganov A.V. On the Chaplygin sphere in a magnetic field, Regular and Chaotic Dynamics, 2019, vol. 24, issue 6, pp. 739-754.
  16. Burov A.A., Subkhankulov G.I. On the motion of a solid in a magnetic field, Journal of Applied Mathematics and Mechanics, 1986, vol. 50, issue 6, pp. 743-748.
  17. Chaplygin S.A. On motion of heavy rigid body of revolution on horizontal plane, Collection of works: Vol. 1, Moscow-Leningrad: OGIZ, 1948, pp. 57-75.
  18. Jaafar R., Chudnovsky E.M., Garanin D.A. Dynamics of the Einstein-de Haas effect: Application to a magnetic cantilever, Physical Review B, 2009, vol. 79, issue 10, 104410.
  19. Garanin D.A., Chudnovsky E.M. Angular momentum in spin-phonon processes, Physical Review B, 2015, vol. 92, issue 2, 024421.
  20. Cushman R. Routh's sphere, Reports on Mathematical Physics, 1998, vol. 42, issue 1-2, pp. 47-70.
  21. Einstein A. Experimenteller nachweis der ampèreschen molekularströme, Naturwissenschaften, 1915, vol. 3, issue 19, pp. 237-238.
  22. Hildebrandt A.F. Magnetic field of a rotating superconductor, Physical Review Letters, 1964, vol. 12, issue 8, pp. 190-191.
  23. Hirsch J.E. Moment of inertia of superconductors, Physics Letters A, 2019, vol. 383, issue 1, pp. 83-90.
  24. Hu S., Santoprete M. Suslov problem with the Clebsch-Tisserand potential, Regular and Chaotic Dynamics, 2018, vol. 23, issue 2, pp. 193-211.
  25. Felderhof B.U. Self-propulsion of a spherical electric or magnetic microbot in a polar viscous fluid, Physical Review E, 2015, vol. 91, issue 2, 023014.
  26. Grioli G. Moto attorno al baricentro di un giroscopio soggetto a forze di potenza nulla, Univ. Roma Ist. Naz. Alta Mat. Rend. Mat. e Appl. (5), 1947, vol. 6, pp. 439-463.
  27. Grioli G. Sul moto di un corpo rigido asimmetrico soggetto a forze di potenza nulla, Rendiconti del Seminario Matematico della Università di Padova, 1957, vol. 27, pp. 90-102.
  28. Goldstein H. The classical motion of a rigid charged body in a magnetic field, American Journal of Physics, 1951, vol. 19, issue 2, pp. 100-109.
  29. Jellet J.H. A treatise on the theory of friction, London: MacMillan, 1872.
  30. Kikoin I.K., Gubar S.W. Gyromagnetic effects in super conductors, J. Phys. USSR, 1940, vol. 3, pp. 333-354.
  31. Kilin A.A., Pivovarova E.N. The rolling motion of a truncated ball without slipping and spinning on a plane, Regular and Chaotic Dynamics, 2017, vol. 22, issue 3, pp. 298-317.
  32. Kilin A.A., Pivovarova E.N. Integrable nonsmooth nonholonomic dynamics of a rubber wheel with sharp edges, Regular and Chaotic Dynamics, 2018, vol. 23, issue 7-8, pp. 887-907.
  33. Kirchhoff G.R. Über die bewegung eines rotationskörpers in einer flüssigkeit, Journal für die Reine und Angewandte Mathematik (Crelles Journal), 1870, vol. 1870, issue 71, pp. 237-262.
  34. Kobrin A.I., Martynenko Yu.G. Motion of a conducting solid body near the center of mass in a slowly varying magnetic field, Soviet Physics Doklady, 1981, vol. 26, no. 12, pp. 1134-1136.
  35. Kozlov V.V. Problem of the rotation of a solid body in a magnetic field, Izv. Akad. Nauk. Mekh. Tverd. Tela, 1985, vol. 20, no. 6, pp. 28-33 (in Russian).
  36. Kroh H.J., Felderhof B.U. Force and torque on a sphere with electric dipole moment moving in a dielectric fluid in the presence of a uniform magnetic field, Physica A: Statistical Mechanics and its Applications, 2000, vol. 280, issue 3-4, pp. 256-265.
  37. Kuznetsov S.P. Regular and chaotic dynamics of a Chaplygin sleigh due to periodic switch of the nonholonomic constraint, Regular and Chaotic Dynamics, 2018, vol. 23, issue 2, pp. 178-192.
  38. Lakshmanan M. The fascinating world of the Landau-Lifshitz-Gilbert equation: an overview, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2011, vol. 369, issue 1939, pp. 1280-1300.
  39. Mentink J.H., Katsnelson M.I., Lemeshko M. Quantum many-body dynamics of the Einstein–de Haas effect, Physical Review B, 2019, vol. 99, issue 6, 064428.
  40. Pry R.H., Lathrop A.L., Houston W.V. Gyromagnetic effect in a superconductor, Physical Review, 1952, vol. 86, issue 6, pp. 905-907.
  41. Rowland H.A. Magnetic effect of electric convection, American Journal of Science, 1878, series 3, vol. 15, pp. 30-38.
  42. Richardson O.W. A mechanical effect accompanying magnetization, Physical Review, 1908, vol. 26, issue 3, pp. 248-253.
  43. Routh E.J. Advanced rigid bodies dynamics, London: MacMillan and Co., 1884.
  44. Samsonov V.A. On the rotation of a body in a magnetic field, Izv. Akad. Nauk. Mekh. Tverd. Tela, 1984, vol. 19, no. 4, pp. 32-34 (in Russian).
  45. Tsiganov A.V. Bäcklund transformations for the nonholonomic Veselova system, Regular and Chaotic Dynamics, 2017, vol. 22, issue 2, pp. 163-179.
  46. Tsiganov A.V. Integrable discretization and deformation of the nonholonomic Chaplygin ball, Regular and Chaotic Dynamics, 2017, vol. 22, issue 4, pp. 353-367.
  47. Tsiganov A.V. On exact discretization of cubic-quintic Duffing oscillator, Journal of Mathematical Physics, 2018, vol. 59, issue 7, 072703.
  48. Tsiganov A.V. Discretization of Hamiltonian systems and intersection theory, Theoretical and Mathematical Physics, 2018, vol. 197, issue 3, pp. 1806-1822.
  49. Tsiganov A.V. Hamiltonization and separation of variables for a Chaplygin ball on a rotating plane, Regular and Chaotic Dynamics, 2019, vol. 24, issue 2, pp. 171-186.
  50. Uhlenbeck G.E., Goudsmit S. Ersetzung der hypothese vom unmechanischen zwang durch eine forderung bezüglich des inneren verhaltens jedes einzelnen elektrons, Naturwissenschaften, 1925, vol. 13, issue 47, pp. 953-954.
  51. Urman Yu.M. Influence of the Barnett-London effect on the motion of a superconducting rotor in a nonuniform magnetic field, Technical Physics, 1998, vol. 43, issue 8, pp. 885-889.
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