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Nonlinear Stability of Riemann Ellipsoids with Symmetric Configurations

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Abstract

Using modern differential geometric methods, we study the relative equilibria for Dirichlet’s model of a self-gravitating fluid mass having at least two equal axes. We show that the only relative equilibria of this type correspond to Riemann ellipsoids for which the angular velocity and vorticity are parallel to the same principal axis of the body configuration. The two solutions found are MacLaurin and transversal spheroids.

The singular reduced energy-momentum method developed in Rodríguez-Olmos (Nonlinearity 19(4):853–877, 2006) is applied to study their nonlinear stability and instability. We found that the transversal spheroids are nonlinearly stable for all eccentricities while for the MacLaurin spheroids, we recover the classical results. Comparisons with other existing results and methods in the literature are also made.

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References

  • Arms, J.M., Marsden, J.E., Moncrief, V.: Symmetry and bifurcations of momentum mappings. Commun. Math. Phys. 78, 455–478 (1980)

    Article  MathSciNet  Google Scholar 

  • Arnold, V.I.: Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluids parfaits. Ann. Inst. Fourier Grenoble 16, 319–361 (1966)

    Google Scholar 

  • Chandrasekhar, S.: Ellipsoidal Figures of Equilibrium. Dover, New York (1987)

    Google Scholar 

  • Ciarlet, P.G.: Mathematical Elasticity. Three Dimensional Elasticity, vol. 1. North-Holland, Amsterdam (1988)

    MATH  Google Scholar 

  • Cohen, H., Muncaster, R.G.: The Theory of Pseudo-Rigid Bodies. Springer Tracts in Natural Philosophy, vol. 33. Springer, New York (1988)

    MATH  Google Scholar 

  • Fassò, F., Lewis, D.: Stability properties of Riemann ellipsoids. Arch. Ration. Mech. Anal. 158, 259–292 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Lebovitz, N.R.: On Riemann’s criterion for the stability of liquid ellipsoids. Astrophys. J. 141, 878–885 (1966)

    Article  Google Scholar 

  • Lerman, E., Singer, S.F.: Stability and persistence of relative equilibria at singular values of the moment map. Nonlinearity 11, 1637–1649 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  • Lewis, X.: Bifurcation of liquid drops. Nonlinearity 6, 491–522 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  • Lewis, D., Simo, J.C.: Nonlinear stability of rotating pseudo-rigid bodies. Proc. R. Soc. Lond. Ser. A 427(1873), 281–319 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  • Liapunov, A.: Sur la stabilité des figures ellipsoidales d’équilibrie d’un liquide animé d’un movement de rotation. Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (2) 6(1), 5–116 (1904)

    MathSciNet  Google Scholar 

  • Marsden, J.E.: Lectures on Mechanics. London Mathematical Society Lecture Note Series, vol. 174. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  • Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliffs (1983)

    MATH  Google Scholar 

  • Montaldi, J.: Persistence and stability of relative equilibria. Nonlinearity 10, 449–466 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  • Ortega, J.-P., Ratiu, T.S.: Stability of Hamiltonian relative equilibria. Nonlinearity 12, 693–720 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Patrick, G.W.: Relative equilibria in Hamiltonian systems: the dynamic interpretation of nonlinear stability on a reduced phase space. J. Geom. Phys. 9, 111–119 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  • Patrick, G.W., Roberts, M., Wulff, C.: Stability of Poisson and Hamiltonian relative equilibria by energy methods. Arch. Ration. Mech. Anal. 174, 301–344 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Perlmutter, M., Rodríguez-Olmos, M., Sousa-Dias, M.E.: The symplectic normal space of a cotangent-lifted action. Differ. Geom. Appl. 26(3), 277–297 (2008)

    Article  MATH  Google Scholar 

  • Poincaré, H.: Sur l’equilibre d’une masse fluide animée d’un mouvement de rotation. Acta Math. 7, 259–380 (1885)

    Article  MathSciNet  Google Scholar 

  • Riemann, B.: Ein Beitrag zu den Untersuchungen über die Bewegung cines flüssigen gleichartigen Ellipsoides. Abh. d. Königl. Gesell. der Wis. zu Göttingen 9, 168–197 (1861)

    Google Scholar 

  • Roberts, M., Sousa-Dias, M.E.: Symmetries of Riemann ellipsoids. Resen. Inst. Mat. Estat. Univ. Sao Paulo 4(2), 183–221 (1999)

    MATH  MathSciNet  Google Scholar 

  • Rodríguez-Olmos, M.: Stability of relative equilibria with singular momentum values in simple mechanical systems. Nonlinearity 19(4), 853–877 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Rodríguez-Olmos, M., Sousa-Dias, M.E.: Symmetries of relative equilibria for simple mechanical systems. In: SPT 2002: Symmetry and Perturbation Theory (Cala Gonone), pp. 221–230. World Scientific, Singapore (2002)

    Google Scholar 

  • Rosensteel, G.: Rapidly rotating nuclei as Riemann ellipsoids. Ann. Phys. 186, 230–291 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  • Rosensteel, G.: Geometric quantization of Riemann ellipsoids. In: Group Theoretical Methods in Physics, Varna, 1987. Lecture Notes in Phys., vol. 313, pp. 253–260. Springer, Berlin (1998)

    Chapter  Google Scholar 

  • Rosensteel, G.: Gauge theory of Riemann ellipsoids. J. Phys. A 34(13), L169–L178 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Simo, J.C., Lewis, D., Marsden, J.E.: Stability of relative equilibria. Part I: The reduced energy-momentum method. Arch. Ration. Mech. Anal. 115, 15–59 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  • Todhunter, M.A.: History of the Theories of Attraction and the Figure of the Earth, vols. I and II. Macmillan and CO., London (1873)

    Google Scholar 

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Authors and Affiliations

  1. Section de Mathématiques, Ecole Polytechnique Fédérale de Lausanne (EPFL), 1015, Lausanne, Switzerland

    Miguel Rodríguez-Olmos

  2. Dep. Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049–001, Lisbon, Portugal

    M. Esmeralda Sousa-Dias

Authors
  1. Miguel Rodríguez-Olmos
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  2. M. Esmeralda Sousa-Dias
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Correspondence to Miguel Rodríguez-Olmos.

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Communicated by: Ratiu.

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Rodríguez-Olmos, M., Sousa-Dias, M.E. Nonlinear Stability of Riemann Ellipsoids with Symmetric Configurations. J Nonlinear Sci 19, 179–219 (2009). https://doi.org/10.1007/s00332-008-9032-z

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  • DOI: https://doi.org/10.1007/s00332-008-9032-z

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