Kovalevskaya top and attitude dynamics of magnetized satellites in equatorial orbits

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Abstract

The article gives a visual illustration of the dynamics of a heavy rigid body around a fixed point in the case of S.V. Kovalevskaya, which arises in the framework of applied problems of space flight. The motion of a rigid body in the case of S.V. Kovalevskaya (the Kovalevskaya top) is equivalent to the dynamics of the attitude of a magnetized satellite around its centre of mass during orbital motion along equatorial circular orbits. The perturbed motion of the magnetized satellite is considered at small deviations from the conditions of the Kovalevskay top, including a small dynamic asymmetry of the satellite, as well as small variations in the magnitude of the external magnetic moment due to weak ellipticity or non-equatoriality of the orbits.

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About the authors

A. V. Doroshin

Samara National Research University

Author for correspondence.
Email: doran@inbox.ru
Russian Federation, Samara

V. S. Aslanov

Samara National Research University

Email: aslanov_vs@mail.ru
Russian Federation, Samara

References

  1. Sophie Kowalevski. Sur le probleme de la rotation d'un corps solide autour d'un point fixe // Acta Math., 1889, vol. 12, pp. 177–232.
  2. Golubev V.V. Lectures on Integrating Equations of Motion of a Heavy Rigid Body around a Fixed Point. Moscow;Leningrad: Gostekhizdat, 1953. 287 p. (in Russian)
  3. Arkhangelsky Yu.A. 1977 Analytical Dynamics of a Rigid Body. Moscow: Nauka, 1977. 328 p. (in Russian)
  4. Kozlov V.V. Dynamic systems arising on the invariant tori of the Kowalewska problem // JAMM, 1975, vol. 39, no. 1, pp. 20–25.
  5. Veselov A.P., Novikov S.P. Poisson brackets and complex tori // Tr. Mat. Inst. Steklov, 1984, vol. 165, pp. 49–61.
  6. Reiman A.G., Semenov-Tyan-Shanskii M.A. Lax representation with a spectral parameter for the kowalevski top and its generalizations // Funct. Anal& Its Appl., 1988, vol. 22, no. 2, pp. 158–160.
  7. Kharlamov M.P. Bifurcation of common levels of first integrals of the Kovalevskaya problem // JAMM, 1983, vol. 47, no. 6, pp. 737–743.
  8. Dullin H.R., Juhnke M., Richter P.H. Action integrals and energy surfaces of the Kovalevskaya top // Int. J. of Bifurc.&Chaos, 1994, vol. 4, no. 06, pp. 1535–1562.
  9. Komarov I.V. A generalization of the Kovalevskaya top // Phys. Lett., 1987, vol. 123, no. 1, pp. 14–15.
  10. Yehia H. New integrable cases in the dynamics of rigid bodies // Mech. Res. Commun., 1986, vol. 13, no. 3, pp. 169–172.
  11. Bogoyavlenskii O.I. Integrable Euler equations on Lie algebras arising in problems of mathematical physics // Izv. RAN. Ser. Matem., 1984, vol. 48, no. 5, pp. 883–938.
  12. Doroshin A.V. Homoclinic solutions and motion chaotization in attitude dynamics of a multi-spin spacecraft // Commun. in Nonlin. Sci.&Numer. Simul., 2014, vol. 19, no. 7, pp. 2528–2552.
  13. Richter P.H., Dullin H.R., Wittek A. Kovalevskaya top // Publ. Wiss. Film. Tech. Wiss. /Naturw, 1997, vol. 13, pp. 33–96.
  14. Borisov A.V., Mamaev I.S. Rigid Body Dynamics. v. 52. Walter de Gruyter GmbH&Co KG, 2018. 521 p.

Supplementary files

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2. Fig. 1. The structure of the ideal geomagnetic field and the change in the magnetic induction vector along satellite orbits

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3. Fig. 2. Dynamic conditions for the implementation of the motion of a heavy rigid body in the case of S.V. Kovalevskaya (a) and similar conditions for the motion of a magnetized satellite in equatorial circular orbits in a geomagnetic field with constant induction Borb (b)

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4. Fig. 3. Time dependences of dynamic parameters in the unperturbed case of S.V. Kovalevskaya under conditions typical for the orbital dynamics of a small spacecraft with a magnetic control system: 1 – dependence G(t) [kg∙m2/s], 2 – dependence L(t) [×10–1 kg∙m2/s], 3 – dependence g(t)∙[×103∙rad], 4 – dependence l(t) [rad], 5 – dependence h(t) [10–1 rad]

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5. Fig. 4. Phase portrait (Poincaré section at g* = 0) {l, L/G} of the unperturbed case of S.V. Kovalevskaya under conditions characteristic of the orbital dynamics of a small spacecraft with a magnetic control system

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6. Fig. 5. Poincare section {l, L/G} (g* = 0) of the unperturbed case

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7. Fig. 6. Poincare section {l, L/G} (g* = 0) of the unperturbed case

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8. Fig. 7. Poincare section {l, L/G} (g* = var) of the unperturbed case

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9. Fig. 8. Poincare section {l, L/G} (g* = var) of the unperturbed case

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10. Fig. 9. Formation of an album of Poincaré sections when cutting volumetric phase portraits with “planes” g = g*

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11. Fig. 10

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12. Fig. 11

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13. Fig. 12. Poincare section {l, L/G} with dynamic asymmetry (B > A)

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14. Fig. 13. Poincare section {l, L/G} and polody ellipsoid with dynamic asymmetry in a strong magnetic field

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15. Fig. 14

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16. Fig. 15

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17. Fig. 16. Using Poinsot's geometric interpretation based on polody ellipsoids recalculated from Poincaré sections

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