The Notion of Dynamical Stability via the Example of Motion of a Rigid Body with a Fixed Point

  • Felix V. Dolzhansky
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 103)


The scale of vortex motions ranges from regular laboratory flows to irregular leakage of water from the tap, to turbulent fluid flows in industrial machines and fast rivers, to circulations in the ocean, in planetary atmospheres, on the stars and even in the formation of galaxies. Emergence of such flows is usually associated with the loss of stability of the so-called primary flow. Its configuration is specified by the initial conditions in case of an ideal fluid or by external energy sources that sustain the motion of a viscous fluid.

The known mechanisms of instability are very few. For instance, the global atmospheric and oceanic currents are mainly formed under the influence of barotropic instability (i.e., due to the presence of horizontal shear of velocity), of baroclinic or convective instability caused by the excess of potential energy of a stratified fluid flow, and of orographic instability evoked by the underlying surface topography. It is also possible that the resonant interaction of planetary waves mentioned in Part II and so-called parametric instability may play a certain role here.


Lyapunov Function Coriolis Force Planetary Wave Resonant Interaction External Energy Source 
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  1. V.I. Arnold, On conditions for nonlinear stability of plane stationary curvilinear flows of an ideal fluid. Dokl. Math., Vol. 162, No. 5, 1965. Google Scholar
  2. L.D. Landau and E.M. Lifschitz, Mechanics, Nauka, GRFML, Moscow, 1973 (in English: 3rd edn, Elsevier Sci., 1976). Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Felix V. Dolzhansky
    • 1
  1. 1.

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