Skip to main content
SpringerLink
Log in

The Hamiltonian dynamics of self-gravitating liquid and gas ellipsoids

  • Research Articles
  • Published:
Regular and Chaotic Dynamics Aims and scope Submit manuscript

Abstract

The dynamics of self-gravitating liquid and gas ellipsoids is considered. A literary survey and authors’ original results obtained using modern techniques of nonlinear dynamics are presented. Strict Lagrangian and Hamiltonian formulations of the equations of motion are given; in particular, a Hamiltonian formalism based on Lie algebras is described. Problems related to nonintegrability and chaos are formulated and analyzed. All the known integrability cases are classified, and the most natural hypotheses on the nonintegrability of the equations of motion in the general case are presented. The results of numerical simulations are described. They, on the one hand, demonstrate a chaotic behavior of the system and, on the other hand, can in many cases serve as a numerical proof of the nonintegrability (the method of transversally intersecting separatrices).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chandrasekhar S., Ellipsoidal Figures of Equilibrium, New Haven, London: Yale University Press, 1969.

    MATH  Google Scholar 

  2. Dirichlet, G. L., Untersuchungen über ein Problem der Hydrodynamik (Aus dessen Nachlass hergestellt von Herrn R. Dedekind zu Zürich), J. reine angew. Math. (Crelle’s Journal), 1861, Bd. 58, S. 181–216.

    MATH  Google Scholar 

  3. Riemann, B., Ein Beitrag zu den Untersuchungen über die Bewegung einer flüssigen gleichartigen Ellipsoïdes, Abh. d. Königl. Gesell. der Wiss. zu Göttingen, 1861.

  4. Jeans, J.H., Problems of Cosmogony and Stellar Dynamics, Cambridge University Press, 1919.

  5. Roche, E., Essai sur la Constitution et l’Origine du Système solaire, Aca. de Montpellier Section des Sciences, 1873, vol. 8, p. 235.

    Google Scholar 

  6. Dirichlet, G. L., Untersuchungen über ein Problem der Hydrodynamik, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen (Mathematisch-Physikalische Klasse), Jg. 1857, No. 14, Aug. 10. S. 203–207 (Dirichlet’s Werke, Bd. 2, S. 28).

  7. Helmholtz, H., Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen, J. reine angew. Math., 1858, B. 55, S. 25–55. Reprinted in: Wissenschaftliche Abhandlungen von Hermann Helmholtz, I, Barth, Leipzig, 1882, S. 101–134.

    MATH  Google Scholar 

  8. Klein, F., Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert (German) [Lectures on the Development of Mathematics in the 19th Century], Berlin-New York: Springer-Verlag, 1979.

    Google Scholar 

  9. Dedekind, R., Zusatz zu der vorstehenden Abhandlung, J. reine angew. Math. (Crelle’s Journal), 1861, Bd. 58, S. 217–228.

    MATH  Google Scholar 

  10. Brioschi, F., Développements relatifs au § 3 des Recherches de Dirichlet sur un problème d’Hydrodynamique, vol. 58, pag. 181 et suivantes de ce Journal, J. reine angew. Math. (Crelle’s Journal), 1861, Bd. 59, S. 63–73.

    MATH  Google Scholar 

  11. Kirchhoff, G., Vorlesungen über mathematische Physik. Mechanik, Leipzig: Teubner, 1876.

    Google Scholar 

  12. Padova, E., Sul moto di un ellissoide fluido ed omogeneo, Annali della Scuola Normale Superiore di Pisa, t. 1, 1871, p. 1–87.

    MathSciNet  Google Scholar 

  13. Lipschitz, R., Reduction der Bewegung eines flüssigen homogenen Ellipsoids auf das Variationsproblem eines einfachen Integrals, und Bestimmung der Bewegung für den Grenzfall eines unendlichen elliptischen Cylinders, J. reine angew. Math. (Crelle’s Journal), 1874, Bd. 78, S. 245–272.

    Article  Google Scholar 

  14. Betti, E., Sopra i moti che conservano la figura ellissoidale a una massa fluida eterogenea, Annali di Matematica Pura ed Applicata, Serie II, 1881, vol. X, pp. 173–187.

    Google Scholar 

  15. Tedone, O., Il moto di un ellissoide fluido secondo l’ipotesi di Dirichlet, Annali della Scuola Normale Superiore di Pisa, 1895, t. 7, pp. I-IV+1–100.

    MathSciNet  Google Scholar 

  16. Basset, A., A Treatise on Hydrodynamics: With Numerous Examples, Vol. II, Ch. 15., Cambridge: Deighton, Bell and Co., 1888.

    MATH  Google Scholar 

  17. Lamb, H., Hydrodynamics, New York: Dover Publications, 1932.

    MATH  Google Scholar 

  18. Thomson, W. and Tait, P.G., Treatise on Natural Philosophy, Cambridge University Press, Part II, 1912 (first edition 1883).

  19. Routh, E. J., A Treatise on Analytical Statics, Cambridge: Cambridge University Press, 1922, Vol. 2.

    Google Scholar 

  20. Appell, P., Traité de Mécanique rationnelle, tome IV: Figures d’équilibre d’une masse liquide homogène en rotation, 2 ed., Paris: Gauthier-Villars, 1932 (IV-1), 1937 (IV-2).

    Google Scholar 

  21. Lyttleton, R.A., The Stability of Rotating Liquid Masses, Cambridge: Cambridge Univ. Press, 1953.

    MATH  Google Scholar 

  22. Basset, A., On the Motion of a Liquid Ellipsoid under the Influence of its Own Attraction, Proc. London Math. Soc., 1885, vol. s1–17, no. 1, pp. 255–262.

    Article  Google Scholar 

  23. Basset, A., On the Stability of a Liquid Ellipsoid which is Rotating about a Principal Axis under the Influence of its Own Attraction, Proc. London Math. Soc., 1887, s1–19, pp. 46–56.

    Article  Google Scholar 

  24. Basset, A., On the Steady Motion of an Annular Mass of Rotating Liquid, Amer. J. Math., 1889, vol. 11, no. 2, pp. 172–181.

    Article  MathSciNet  Google Scholar 

  25. Duhem, M.P., Sur la stabilite de l’équilibre relatif d’une masse fluide animée d’un mouvement de rotation, J. de Math. Pures et Appl., 1905, vol. 7, ser. 5, pp. 331–350.

    Google Scholar 

  26. Hagen, J., Ueber die Stabilitat des Gleichgewichtes einer auf einem dreiaxigen Ellipsoid mit kleinen Excentricitaten ausgebreiteten Flussigkeit, Zeitschrift für Mathematik und Physik, 1877, vol. 22, pp. 65–86.

    Google Scholar 

  27. Hicks, W. M., On the Motion of a Mass of Liquid under its Own Attraction, when the Initial Form is an Ellipsoid, Proc. Camb. Phil. Soc., 1883, Vol. IV, Pt. VI, pp. 1–4.

    Google Scholar 

  28. Hill, M. J. M., Note on the Motion of a Fluid Ellipsoid under its Own Attraction, Proc. London Math. Soc., 1891, s1–23, pp. 88–95.

    Article  Google Scholar 

  29. Love, A. E.H., On the Motion of a Liquid Elliptic Cylinder under its Own Attraction, Quart. J. of Pure and Appl. Math., 1889, vol. 23, pp. 153–158.

    Google Scholar 

  30. Love, A.E.H., The Oscillations of a Mass of Gravitating Liquid in the Form of an Elliptic Cylinder which Rotates as if Rigid about its Axis, Quart. J. of Pure and Appl. Math., n.d., pp. 158–165.

  31. Lyapunov, A.M., Collected Works, Collected Works, Vol. 5, Moscow: Izd. Akad. Nauk, 1965.

    Google Scholar 

  32. Lyapunov, A.M., On Certain Series of the Figures of Equilibrium of Non-homogeneous rotating fluid, in Collected Works, Moscow: Izd. Akad. Nauk., 1965, pp. 7–378. See also Steklov, V.A., Post-mortem Lyapunov’s Works on the Figures of Equilibrium of Non-homogeneous Rotating Fluid, ibid, pp. 7–378.

    Google Scholar 

  33. Poincaré, H., Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation, Acta Math., vol. 7, 1885, pp. 259–380.

    Article  MathSciNet  Google Scholar 

  34. Poincaré, H., Figures d’équilibre d’une masse fluide (Leçons professées à la Sorbonne en 1900), Paris: Gauthier-Villars, 1902.

    Google Scholar 

  35. Darwin, G.H., On the Figure and Stability of a Liquid Satellite, Phil. Trans. Roy. Soc. London, 1906, vol. 206, pp. 161–248; see also Scientific Papers, vol. 3, Cambridge University Press, 1910, p. 436.

    Article  Google Scholar 

  36. Sretenskij, L.N., The Theory of the Figures of Equilibrium of the Rotating Flui Mass, 1938, Uspekhi mat. nauk, no. 5, pp. 187–230 (in Russian).

  37. Giesen, A., Über die Stabilität des Gleichgewichtes einer nur der Gravitation unterworfenen Fliissigkeit, Jahres-Bericht fiber die hohere Schule in Opladen, Bonn, 1872–1873.

  38. Bryan, G. H., On the Stability of a Rotating Spheroid of Perfect Liquid, Proc. Roy. Soc. London, 1889–1890, vol. 47, pp. 367–376.

    Article  Google Scholar 

  39. Liouville, J., Formules générales relatives à la question de la Stabilité de l’équilibre d’une masse liquide homog`ene douée d’un mouvement de rotation autour d’un axe, J. Math. Pures Appl., 1855, vol. 20, pp. 164–184.

    Google Scholar 

  40. Dyson, F. W., The Potentials of Ellipsoids of Variable Densities, Quart. J. Pure Appl. Math., 1891, vol. 25, pp. 259–288.

    Google Scholar 

  41. Ferrers, N.M., On the Potentials, Ellipsoids, Ellipsoidal Shells, Elliptic Laminae, and Elliptic Rings, of Variable Densities, Quart. J. Pure Appl. Math., 1875, vol. 14, pp. 1–23.

    Google Scholar 

  42. Volterra, V., Sur la Stratification d’une Masse Fluide en Équilibre, Acta Math., 1903, vol. 27, no. 1, pp. 105–124.

    Article  MATH  MathSciNet  Google Scholar 

  43. Lichtenstein, L., Gleichgewichtsfiguren rotietender Flüssigkeiten, Berlin: Springer-Verlag, 1933.

    Google Scholar 

  44. Veronnet, A., Rotation de l’ellipsoide hétérogène et figure exacte, J. de Math. Pures et Appl., 1912, vol. 8, ser. 6, pp. 331–463.

    Google Scholar 

  45. Pizzetti, P., Principii della teoria meccanica della figura dei pianeti, Pisa, 1912.

  46. Cisneros, J.U., Martinez F. J., and Montalvo J. D., On the Stability of a Self-Gravitating Inhomogeneous Fluid in the form of Two Confocal Spheroids Rotating with Different Angular Velocities, Rev. Mexicana Astron. Astrof., 2000, vol. 36, pp. 185–210.

    Google Scholar 

  47. Cisneros, J.U., Martinez F. J., and Montalvo J. D., On the Stability of a Self-Gravitating Inhomogeneous Fluid in the Form of Two Confocal Ellipsoids Carrying Dedekind-type Internal Currents, Rev. Mexicana Astron. Astrof., 2004, vol. 40, pp. 167–180.

    Google Scholar 

  48. Martinez, F. J., Cisneros, J., and Montalvo, D., On Equilibrium Figures for Ideal Fluids in the Form of Confocal Ellipsoids Rotating with Common Angular Velocity, Rev. Mexicana Astron. Astrof., 1990, vol. 20, pp. 15–22. See also: Erratum, ibid., pp. 153–154.

    Google Scholar 

  49. Ipser, J.R. and Managan, R.A., On the Existence and Structure of Inhomogeneous Analogs of the Dedekind and Jacobi Ellipsoids, Astrophys. J., 1981, vol. 250, pp. 362–372.

    Article  MathSciNet  Google Scholar 

  50. Lebovitz, N.R., On the Principle of the Exchange of Stabilities. I. The Roche Ellipsoids, Astrophys. J., 1963, vol. 138, pp. 1214–1217.

    Article  Google Scholar 

  51. Lebovitz, N.R., The Mathematical Development of the Classical Ellipsoids, Intern. Jour. Engineering Science, 1998, vol. 36, no. 12, pp. 1407–1420.

    Article  MathSciNet  Google Scholar 

  52. Bohr, N. and Wheeler, J., The Mechanism of Nuclear Fission, Phys. Rev., 1939, vol. 56, pp. 426–450.

    Article  MATH  Google Scholar 

  53. Rosensteel, G. and Tran, H.Q., Hamiltonian Dynamics of Self-gravitating Ellipsoids, The Astrophysical Journal, 1991, vol. 366, pp. 30–37.

    Article  Google Scholar 

  54. Rosensteel, G., Gauge Theory of Riemann Ellipsoids, J. Phys. A: Math. Gen., 2001, vol. 34, L1–L10.

    Article  MathSciNet  Google Scholar 

  55. Graber, J.L. and Rosensteel, G., Circulation of a Triaxial, Charged Ellipsoidal Droplet, Phys. Rev. C, 2002, vol. 66, 034309.

  56. Rosenkilde, C.E., Stability of Axisymmetric Figures of Equilibrium of a Rotating Charged Liquid Drop, J. Math. Phys., 1967, vol. 8, no. 1, pp. 98–118.

    Article  Google Scholar 

  57. Sudakov, S. N., On Oscillation of Rotating liquid Ellipsoids with Variable Density, Mekh. Tverd. Tela, Donetsk, 2002, no. 32, pp. 208–217 (in Russian).

  58. Eriguchi, Y. and Muller, E., A General Computational Method for Obtaining Equilibria of Selfgravitating and Rotating Gases, Astron. Astrophys., 1985, vol. 146, pp. 260–268.

    MATH  Google Scholar 

  59. Narlikar, V.V. and Larmor, J., The Kelvin-Poincare Problem of Stellar Evolution, Proc. Roy. Soc. London Ser. A, 1934, vol. 144, no. 851, pp. 28–46.

    Article  MATH  Google Scholar 

  60. Fassó, F. and Lewis, D., Stability Properties of the Riemann Ellipsoids, Arch. Rational Mech. Anal., 2001, vol. 158, pp. 259–292.

    Article  MATH  Google Scholar 

  61. Holm, D.D., Magnetic Tornadoes:Three-Dimensional Affine Motions in Ideal Magnetohydrodynamics, Phys. D, 1983, vol. 8, pp. 170–182.

    Article  MATH  MathSciNet  Google Scholar 

  62. Biello, J.A., Lebovitz, N.R., and Morison, P.J., Hamiltonian Reduction of Incompressible Fluid Ellipsoids, Preprint, http://people.cs.uchicago.edu/lebovitz/hamred.pdf.

  63. Borisov, A.V. and Mamaev, I.S., Rigid Body Dynamics, Moscow-Izhevsk: Inst. Comp. Sci., RCD, 2005 (in Russian).

    MATH  Google Scholar 

  64. Roche, E., Mémoire sur la figure d’une masse fluide, soumise a l’attraction d’un point éloigné, Acad. des Sci. de Montpellier, 1849–1850 t. 1, pp. 243–262, 333–348; 1852, t. 2, pag. 21.

  65. Stekloff, W., Problème du mouvement d’une masse fluide incompressible de la forme ellipsoïdale les parties s’attirent suivant la loi de Newton, Annales scientifiques de l’ É.N.S. 3e série, 1908, t. 25, pp. 469–528.

  66. Stekloff, W., Problème du mouvement d’une masse fluide incompressible de la forme ellipsoïdale les parties s’attirent suivant la loi de Newton (Suite.), Annales scientifiques de l’ É.N.S. 3e série, 1909, t. 26, pp. 275–336.

  67. Marshalek, E. R., An overlooked figure of equilibrium of a rotating ellipsoidal self-gravitating fluid and the Riemann theorem, Phys. Fluids, 1996, vol. 8, no. 12, pp. 3414–3422.

    Article  MATH  MathSciNet  Google Scholar 

  68. Ziglin, S.L., Personal communication.

  69. Roberts, W., Application des coordonnées elliptiques `a la recherche des surfaces orthogonales J. reine angew. Math. (Crelle’s Journal), 1863, vol. 62, pp. 50–60.

    MATH  Google Scholar 

  70. Wangerin, A., Uber ein dreifach orthogonales Flachensystem, gebildet aus gewissen Flächen vierter Ordnung, J. reine angew. Math. (Crelle’s Journal), 1876, vol. 82, pp. 145–157.

    Google Scholar 

  71. Schläfli, L., On the Distribution of Surfaces of the Third Order into Species, in Reference to the Absence or Presence of Singular Points, and the Reality of Their Lines, Philos. Trans. Roy. Soc. London, 1863, vol. 153, pp. 193–241.

    Article  Google Scholar 

  72. Cayley A., A Memoir on Cubic Surfaces, Philos. Trans. Roy. Soc. London, 1869, vol. 159, pp. 231–326.

    Article  Google Scholar 

  73. Kozlov, V.V., Topology of Real Algebraic Curves, Funktsional. Anal. i Prilozhen., 2008, vol. 42, no. 2, pp. 23–27 [Functional Analysis and Its Applications, 2008, vol. 42, no. 2, pp. 98–102].

    Article  MathSciNet  Google Scholar 

  74. Ovsyannikov, L.V., A New Solution of the Equations of Hydrodynamics, Dokl. Akad. Nauk SSSR (N.S.), 1956, vol. 111, pp. 47–49 (in Russian).

    MATH  MathSciNet  Google Scholar 

  75. Lynden-Bell, D., On the Gravitational Collapse of a Cold Rotating Gas Cloud, Proc. Camb. Phys. Soc., 1962, vol. 58, pp. 709–711.

    Article  Google Scholar 

  76. Zel’dovich, Ya.B., Newtonian and Einsteinian Motion of Homogeneous Matter, Astronom. Zh., 1964, vol. 41, no. 5, pp. 872–883 [Soviet Astronomy, 1964, vol. 8, no. 5].

    Google Scholar 

  77. Dyson, F. J., Dynamics of a Spinning Gas Cloud, J. Math. Mech., 1968, vol. 18, no. 1, pp. 91–101.

    MATH  Google Scholar 

  78. Fujimoto, F., Gravitational Collapse of Rotating Gaseous Ellipsoids, Astrophys. J., 1968, vol. 152, no. 2 pp. 523–536.

    Article  MathSciNet  Google Scholar 

  79. Rossner, L. F., The Finite-amplitude Oscillations of the Maclaurin Spheroids, Astrophys. J., 1967, vol. 149, pp. 145–168.

    Article  MATH  Google Scholar 

  80. Anisimov, S.I. and Lysikov, Iu.I, Expansion of a Gas Cloud in Vacuum, Prikl. mat. mekh., 1970, vol. 34, no. 5, pp. 926–929 [J. Appl. Math. Mech., 1970, vol. 34, no. 5, pp. 882–885].

    Google Scholar 

  81. Bogoyavlenskij, O.I., Dynamics of a gravitating gaseous ellipsoid, Prikl. mat. mekh., 1976, vol. 40, no. 2, pp. 270–280 [J. Appl. Math. Mech., 1976, vol. 40, no. 2, pp. 246–256].

    Google Scholar 

  82. Jacobi, C.G. J., Problema trium corporum mutuis attractionibus cubis distantiarum inverse proportionalibus recta linea se moventium, Gesammelte Werke, Vol. 4, Berlin: Reimer, 1886. S. 531–539.

    Google Scholar 

  83. Gaffet, B., Expanding Gas Clouds of Ellipsoidal Shape: New Exact Solutions, J. Fluid Mech., 1996, vol. 325, pp. 113–144.

    Article  MATH  MathSciNet  Google Scholar 

  84. Gaffet, B., Sprinning Gas without Vorticity: the Two Missing Integrals, J. Phys. A: Math. Gen., 2001, vol. 34, pp. 2087–2095.

    Article  MATH  MathSciNet  Google Scholar 

  85. Gaffet, B., Sprinning Gas Clouds: Liouville Integrability, J. Phys. A: Math. Gen., 2001, vol. 34, pp. 2097–2109.

    Article  MATH  MathSciNet  Google Scholar 

  86. Lidov, M.L., Exact Solutions of the Equations of One-dimensional Unsteady Motion of a Gas, Taking Account of the Forces of Newtonian Attraction, Doklady Akad. Nauk SSSR (N.S.), 1951, vol. 97, pp. 409–410 (in Russian).

    MathSciNet  Google Scholar 

  87. Nemchinov, I.V., Expansion of a Tri-axial Gas Ellipsoid in a Regular Behavior, Prikl. mat. mekh., 1965, vol. 29, no. 1, pp. 134–140 [J. Appl. Math. Mech., 1965, vol. 29, no. 1, pp. 143–150].

    Google Scholar 

  88. Deryabin, S.L., One-Dimension Escape of Self-Gravitating Ideal Gas Into Vacuum, Computational technolgies, 2003, vol. 8, no. 4, pp. 32–44.

    MATH  Google Scholar 

  89. Landau, L.D. and Lifshits, E.M., Theoretical physics, Vol. VI. Hydrodynamics, Moscow: Nauka, 1986.

    Google Scholar 

  90. Albouy, A. and Chenciner, A. Le problème des n Corps et les Distances Mutuelles, Invent. Math., 1998, vol. 131, pp. 151–184.

    Article  MATH  MathSciNet  Google Scholar 

  91. Gaffet, B., Analytical Methods for the Hydrodynamical Evolution of Supernova Remnants. II - Arbitrary Form of Boundary Conditions, Astrophysical Journal, Part 1, vol. 249, 1981, pp. 761–786.

    Article  Google Scholar 

  92. Gaffet, B., Two Hidden Symmetries of the Equations of Ideal Gas Dynamics, and the General Solution in a Case of Nonuniform Entropy Distribution, J. Fluid Mech., 1983, vol. 134, p.179–194.

    Article  MATH  MathSciNet  Google Scholar 

  93. Gaffet, B., SU(3) Symmetry of the Equations of Unidimensional Gas Flow, with Arbitrary Entropy Distribution, J. Math. Phys., 1984, vol. 25, no. 2, pp. 245–255.

    Article  MATH  MathSciNet  Google Scholar 

  94. Perelomov, A., Integrable Systems of Classical Mechanics and Lie Algebras, Basel: Birkhäser, 1990.

    Google Scholar 

  95. Wojciechowski, S., An Integrable Marriage of the Euler Equations with the Calogero-Moser System, Phys. Lett. A, 1985, vol. 111, no. 3, pp. 101–103.

    Article  MathSciNet  Google Scholar 

  96. Khvedelidze, A. and Mladenov, D., Euler-Calogero-Moser System from SU(2) Yang-Mills Theory, arXiv:hep-th/9906033v3.

  97. Tsiganov, A.V., On an Integrable System Related to a Top and the Toda Lattice, Theor. mat. fiz., 2000, vol. 124, pp. 310–322 (in Russian).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computer Science, Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034, Russia

    A. V. Borisov, A. A. Kilin & I. S. Mamaev

Authors
  1. A. V. Borisov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Kilin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. I. S. Mamaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Borisov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Borisov, A.V., Kilin, A.A. & Mamaev, I.S. The Hamiltonian dynamics of self-gravitating liquid and gas ellipsoids. Regul. Chaot. Dyn. 14, 179–217 (2009). https://doi.org/10.1134/S1560354709020014

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354709020014

MSC2000 numbers

Key words

Search

Navigation