Hydrodynamic Interpretation of the Euler Equations of Motion of a Classical Gyroscope and Their Invariants

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


In 1879, a prominent English hydrodynamist A.G. Greenhill made an observation whose theoretical value was recognized almost a century later. He observed that the Euler equations for a rigid body with a fixed point describe the flow of an ideal homogeneous incompressible fluid (whose equations of motion are also named after Euler) inside a triaxial ellipsoid within the class of linear velocity fields. This discovery was used, in particular, by such classics of science as N.E. Zhukovskii, S.S. Hough, and H. Poincaré to study the motions of solids with cavities filled with a fluid (see Moiseev and Rumyantsev, in The Dynamics of Bodies with Liquid-Filled Cavities 1965).


Angular Velocity Euler Equation Coriolis Force Constant Angular Velocity Rotate Coordinate System 
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  2. F.V. Dolzhansky, On the mechanical prototypes of the fundamental hydrodynamical invariants and slow manifolds, UFN, Vol. 175, No. 12, 2005. Google Scholar
  3. 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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