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).
Keywords:
liquid and gas self-gravitating ellipsoids, integrability, chaotic behavior.
Citation:
A. V. Borisov, I. S. Mamaev, A. A. Kilin, “The Hamiltonian Dynamics of Self-gravitating Liquid and Gas Ellipsoids”, Regul. Chaotic Dyn., 14:2 (2009), 179–217
\Bibitem{BorMamKil09}
\by A. V. Borisov, I. S. Mamaev, A. A. Kilin
\paper The Hamiltonian Dynamics of Self-gravitating Liquid and Gas Ellipsoids
\jour Regul. Chaotic Dyn.
\yr 2009
\vol 14
\issue 2
\pages 179--217
\mathnet{http://mi.mathnet.ru/rcd546}
\crossref{https://doi.org/10.1134/S1560354709020014}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2505424}
\zmath{https://zbmath.org/?q=an:1229.70049}
Linking options:
https://www.mathnet.ru/eng/rcd546
https://www.mathnet.ru/eng/rcd/v14/i2/p179
This publication is cited in the following 20 articles:
Renata Nikonorova, Dilara Siraeva, Yulia Yulmukhametova, “New Exact Solutions with a Linear Velocity Field for the Gas Dynamics Equations for Two Types of State Equations”, Mathematics, 10:1 (2022), 123
Olga P. Stoyanovskaya, Vitaliy V. Grigoryev, Anastasiya N. Suslenkova, Maxim N. Davydov, Nikolay V. Snytnikov, “Two-Phase Gas and Dust Free Expansion: Three-Dimensional Benchmark Problem for CFD Codes”, Fluids, 7:2 (2022), 51
Guo Ya., Hadzic M., Jang J., “Continued Gravitational Collapse For Newtonian Stars”, Arch. Ration. Mech. Anal., 239:1 (2021), 431–552
Fan E., Yuen M., “A Method For Constructing Special Solutions For Multidimensional Generalization of Euler Equations With Coriolis Force”, Chin. J. Phys., 72 (2021), 136–144
Rickard C., “The Vacuum Boundary Problem For the Spherically Symmetric Compressible Euler Equations With Positive Density and Unbounded Entropy”, J. Math. Phys., 62:2 (2021), 021504
Olga Rozanova, Marko Turzynsky, Springer Proceedings in Mathematics & Statistics, 292, Nonlinear Analysis and Boundary Value Problems, 2019, 131
Oleg N. Kirillov, CISM International Centre for Mechanical Sciences, 586, Dynamic Stability and Bifurcation in Nonconservative Mechanics, 2019, 129
E. Yu. Prosviryakov, “Dynamic equilibria of a nonisothermal fluid”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 22:4 (2018), 735–749
G. Rosensteel, N. Sparks, “SU(3) gauge theory of nuclear rotations”, EPL, 119:6 (2017), 62001
C. Ragazzo, L. S. Ruiz, “Dynamics of an isolated, viscoelastic, self-gravitating body”, Celest Mech Dyn Astr, 122:4 (2015), 303
Hugo A. Folonier, Sylvio Ferraz-Mello, Konstantin V. Kholshevnikov, “The flattenings of the layers of rotating planets and satellites deformed by a tidal potential”, Celest Mech Dyn Astr, 122:2 (2015), 183
Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Figures of equilibrium of an inhomogeneous self-gravitating fluid”, Celest. Mech. Dyn. Astr., 122:1 (2015), 1–26
I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Figury ravnovesiya neodnorodnoi samogravitiruyuschei zhidkosti”, Nelineinaya dinam., 10:1 (2014), 73–100
Thierry Combot, “Non-integrability of a Self-gravitating Riemann Liquid Ellipsoid”, Regul. Chaotic Dyn., 18:5 (2013), 497–507
Giorgio Fusco, Piero Negrini, Waldyr M. Oliva, “Stationary Motion of a Self-gravitating Toroidal Incompressible Liquid Layer”, Regul. Chaotic Dyn., 17:5 (2012), 397–416
T. B. Ivanova, “Postroenie bifurkatsionnoi diagrammy i analiz ustoichivosti zhidkogo samogravitiruyuschego ellipticheskogo tsilindra s vnutrennim vrascheniem”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2010, no. 4, 77–86
A. V. Borisov, I. S. Mamaev, T. B. Ivanova, “Ustoichivost zhidkogo samogravitiruyuschego ellipticheskogo tsilindra s vnutrennim vrascheniem”, Nelineinaya dinam., 6:4 (2010), 807–822
B Gaffet, “Spinning gas clouds with precession: a new formulation”, J. Phys. A: Math. Theor., 43:16 (2010), 165207
A. V. Borisov, I. S. Mamaev, “Isomorphisms of geodesic flows on quadrics”, Regul. Chaotic Dyn., 14:4 (2009), 455–465
Citation in format AMSBIB
Contact us: