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Fractal theory of Saturn’s ring. (English. Russian original) Zbl 1337.85001

Proc. Steklov Inst. Math. 291, 87-101 (2015); translation from Tr. Mat. Inst. Steklova 291, 95-111 (2015).
Summary: The true reason, in our opinion, for the partition of Saturn’s ring as well as the rings of other planets into a large number of small subrings is found. This reason is clarified by the Zelikin-Lokutsievskiy-Hildebrand theorem about the fractal structure of solutions to generic piecewise smooth Hamiltonian systems. The instability of the two-dimensional model of the ring with continuous surface density of the distribution of particles is proved both for the Newton and Boltzmann equations. We do not claim that we have solved the problem of stability of Saturn’s ring. We rather put questions and suggest some ideas and means for future research.

MSC:

85A15 Galactic and stellar structure
85A05 Galactic and stellar dynamics
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
70F15 Celestial mechanics
28A80 Fractals
35Q20 Boltzmann equations
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
35Q31 Euler equations
76E20 Stability and instability of geophysical and astrophysical flows
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References:

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