Advertisement

Orbital stability in static axisymmetric fields

  • Gopakumar Mohandas
  • Tobias HeinemannEmail author
  • Martín E. Pessah
Original Article
  • 110 Downloads

Abstract

We investigate the stability of circular orbits in static axisymmetric, but otherwise arbitrary, gravitational and electromagnetic fields. We extend previous studies of this problem to include a toroidal magnetic field. We find that even though the toroidal magnetic field does not alter the location of circular orbits, given by the critical points of the effective potential, it does affect their stability. This is because a circular orbit located at an isolated maximum of the effective potential—which in the absence of a toroidal magnetic field is an unstable configuration—can be rendered stable by a toroidal magnetic field through the phenomenon of gyroscopic stabilization. We find that for any such maximum, gyroscopic stabilization is always possible given a sufficiently strong toroidal magnetic field. We also show that no isolated maxima exist in source-free regions of space. As an example of a force field produced in part by a continuous charge distribution throughout space, we consider a rotating dipolar magnetosphere. We show that in this case a toroidal magnetic field can indeed provide gyroscopic stabilization for positively charged particles in prograde equatorial orbits.

Keywords

Gyroscopic stabilization Magnetic fields Axisymmetry Orbital stability 

Notes

Acknowledgements

We thank Pablo Benítez-Llambay, Luis García-Naranjo and Jihad Touma for insightful comments. We are grateful for the hospitality of the Institute for Advanced Study where part of this work was carried out. We thank the anonymous referees whose inquiries and comments contributed to improving this manuscript. The research leading to these results has received funding from the European Research Council (ERC) under the European Union’s Seventh Framework programme (FP/2007–2013) under ERC Grant Agreement No. 306614.

References

  1. Almaguer, J.A., Hameiri, E., Herrera, J., Holm, D.D.: Lyapunov stability analysis of magnetohydrodynamic plasma equilibria with axisymmetric toroidal flow. Phys. Fluids 31(7), 1930 (1988)ADSCrossRefGoogle Scholar
  2. Arnold, V.I.: Small denominators and problems of stability of motion in classical and celestial mechanics. Russ. Math. Surv. 18(6), 85–191 (1963)MathSciNetCrossRefGoogle Scholar
  3. Bloch, A., Krishnaprasad, P.S., Marsden, J.E., Ratiu, T.: Dissipation Induced Instabilities. Annales de l’Institut Henri Poincaré (C) Analyse Non Linéaire 11(1), 37–90 (1994)ADSMathSciNetCrossRefGoogle Scholar
  4. Bolotin, S., Negrini, P.: Asymptotic solutions of Lagrangian systems with gyroscopic forces. Nonlinear Differ. Equ. Appl. NoDEA 2(4), 417–444 (1995)MathSciNetCrossRefGoogle Scholar
  5. Chetaev, N.G.: The Stability of Motion. Pergamon Press, Oxford (1961)Google Scholar
  6. Dullin, H.R., Horányi, M., Howard, J.E.: Generalizations of the Störmer problem for dust grain orbits. Phys. D Nonlinear Phenom. 171(3), 178–195 (2002)ADSCrossRefGoogle Scholar
  7. Goldreich, P., Julian, W.H.: Pulsar electrodynamics. Astrophys. J. 157(August), 869 (1969)ADSCrossRefGoogle Scholar
  8. Hagedorn, P.: Die Umkehrung der Stabilitätssätze von Lagrange–Dirichlet und Routh. Arch. Ration. Mech. Anal. 42(4), 281–316 (1971)MathSciNetCrossRefGoogle Scholar
  9. Haller, G.: Gyroscopic stability and its loss in systems with two essential coordinates. Int. J. Non-Linear Mech. 27(1), 113–127 (1992)ADSMathSciNetCrossRefGoogle Scholar
  10. Holm, D.D., Marsden, J.E., Ratiu, T., Weinstein, A.: Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123(1–2), 1–116 (1985)ADSMathSciNetCrossRefGoogle Scholar
  11. Howard, J.E.: Stability of relative equilibria in arbitrary axisymmetric gravitational and magnetic fields. Celest. Mech. Dyn. Astron. 74(1), 19–57 (1999)ADSMathSciNetCrossRefGoogle Scholar
  12. Howard, J.E., Horányi, M., Stewart, G.R.: Global dynamics of charged dust particles in planetary magnetospheres. Phys. Rev. Lett. 83(20), 3993–3996 (1999)ADSCrossRefGoogle Scholar
  13. Howard, J.E., Dullin, H.R., Horányi, M.: Stability of halo orbits. Phys. Rev. Lett. 84(15), 3244–7 (2000)ADSCrossRefGoogle Scholar
  14. Jauch, J.M.: Gauge invariance as a consequence of Galilei-invariance for elementary particles. Helv. Phys. Acta 37(3), 284–292 (1964)MathSciNetzbMATHGoogle Scholar
  15. Kolmogorov, A.: On the conservation of conditionally periodic motions under small perturbation of the hamiltonian. Dokl. Akad. Nauk. SSR 98(527), 2–3 (1954)Google Scholar
  16. Krechetnikov, R., Marsden, J.E.: Dissipation-induced instabilities in finite dimensions. Rev. Mod. Phys. 79(2), 519–553 (2007)ADSMathSciNetCrossRefGoogle Scholar
  17. Littlejohn, R.G.: A guiding center Hamiltonian: a new approach. J. Math. Phys. 20(12), 2445–2458 (1979)ADSMathSciNetCrossRefGoogle Scholar
  18. Littlejohn, R.G.: Hamiltonian perturbation theory in noncanonical coordinates. J. Math. Phys. 23(5), 742–747 (1982)ADSMathSciNetCrossRefGoogle Scholar
  19. Lyapunov, A.: Problème général de la stabilité du mouvement. Annales de la Faculté des sciences de Toulouse: Mathématiques 9, 203–474 (1907)MathSciNetCrossRefGoogle Scholar
  20. MacKay, R.S.: Movement of eigenvalues of Hamiltonian equilibria under non-Hamiltonian perturbation. Phys. Lett. A 155(4–5), 266–268 (1991)ADSMathSciNetCrossRefGoogle Scholar
  21. Malkin, J.G.: Theorie der Stabilität einer Bewegung. Akademie, Berlin (1959)CrossRefGoogle Scholar
  22. Marsden, J.E., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5(1), 121–130 (1974)ADSMathSciNetCrossRefGoogle Scholar
  23. Merkin, D.R.: Introduction to the Theory of Stability. Texts in Applied Mathematics, vol. 24. Springer, New York (1996)CrossRefGoogle Scholar
  24. Modesitt, G.E.: Maxwell’s equations in a rotating reference frame. Am. J. Phys. 38(1970), 1487 (1970). http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1970AmJPh..38.1487M&link_type=EJOURNAL%5Cnpapers://ad348f84-eb9b-4b43-b3a1-f1af266abee1/Paper/p4499
  25. Moser, J.: On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen II, 1–20 (1962)MathSciNetzbMATHGoogle Scholar
  26. Moser, J.: Lectures on Hamiltonian Systems: Rigorous and Formal Stability of Orbits about an Oblate Planet; Memoirs of the American Mathematical Society, 81 (1968)Google Scholar
  27. Pars, L.A.: A Treatise on Analytical Dynamics. Heinemann, London (1965)zbMATHGoogle Scholar
  28. Roman, P., Leveille, J.P.: Gauge theories and Galilean symmetry. J. Math. Phys. 15(10), 1760–1767 (1974)ADSMathSciNetCrossRefGoogle Scholar
  29. Routh, E.J.: The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies. Cambridge University Press, Cambridge (1884)Google Scholar
  30. Rumiantsev, V.: On the stability of steady motions. J. Appl. Math. Mech. 30(5), 1090–1103 (1966)MathSciNetCrossRefGoogle Scholar
  31. Rumyantsev, V.V., Sosnitskii, S.P.: The stability of the equilibrium of holonomic conservative systems. J. Appl. Math. Mech. 57(6), 1101–1122 (1993)MathSciNetCrossRefGoogle Scholar
  32. Salvadori, L.: Un’osservazione su di un criterio di stabilita del Routh. Rend. Accad. Sci. Fis. e math. Soc. Nza. Lett. Ed. Arti. Napoli 20(1–2), 269–272 (1953)MathSciNetzbMATHGoogle Scholar
  33. Schiff, L.I.: A question in general relativity. Proc. Natl. Acad. Sci. 25(7), 391–395 (1939).  https://doi.org/10.1073/pnas.25.7.391 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. Thomson, W., Tait, P.G.: Treatise on Natural Philosophy, vol. 1. Cambridge University Press, Cambridge (1883)zbMATHGoogle Scholar
  35. Thyagaraja, A., McClements, KG.: Plasma physics in noninertial frames. Phys. Plasm. 16(9), 092506 (2009). http://link.aip.org/link/PHPAEN/v16/i9/p092506/s1&Agg=doi

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Niels Bohr International AcademyNiels Bohr InstituteCopenhagenDenmark

Personalised recommendations