On Some Results of Investigation of Kirchhoff Equations in Case of a Rigid Body Motion in Fluid
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Some results of analysis of Kirchhoff equations, which describe the motion of a rigid body in the ideal incompressible fluid, are presented. With respect to these equations, a problem is stated to obtain steady-state motions, invariant manifolds of steady-state motions (IMSMs), and to investigate their properties in the aspect of stability and stabilization of motion. Our methods of investigation are based on classical results obtained by Lyapunov . The computer algebra systems (CAS) “Mathematica”, “Maple”, and a software  are used as the tools. Lyapunov’s sufficient stability conditions are derived for some steady-state motions obtained. A problem of optimal stabilization with respect to the first approximation equations is solved for some cases of unstable motion. This paper represents a continuation of our research, the results of which have been reported during CASC’2004 in St. Petersburg .
KeywordsLyapunov Function Invariant Manifold Rigid Body Motion Computer Algebra System Nonlinear Algebraic Equation
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- 1.Lyapunov, A.M.: The general problem of stability of motion. Collected Works. USSR Acad. Sci. Publ., Moscow–Leningrad 2 (1956)Google Scholar
- 2.Irtegov, V.D., Titorenko, T.N.: Computer algebra and investigation of invariant manifolds of mechanical systems. Mathematics and Computers in Simulation. Special Issue: Applications of Computer Algebra in Science, Engineering, Simulation and Special Software 67(1-2), 99–109 (2004)zbMATHMathSciNetGoogle Scholar
- 3.Irtegov, V., Titorenko, T.: On stability of body motions in fluid. In: Proc. Seventh Workshop on Computer Algebra in Scientific Computing, pp. 261–268. Munich Techn. Univ. (2004)Google Scholar
- 4.Dolzhansky, F.V.: On hydrodynamical interpretation of equations of rigid body motion. Izvestiya of Russian Academy of Sciences. Physics of Atmosphere and Ocean 2(13), 201–204 (1977)Google Scholar
- 6.Oden, M.: Rotating Tops: A Course Integrable Systems. Izhevsk, Udmurtiya univ. (1999)Google Scholar
- 7.Lyapunov, A.M.: On Permanent Helical Motions of a Rigid Body in Fluid. Collected Works. USSR Acad. Sci. Publ., Moscow–Leningrad 1, 276–319 (1954)Google Scholar
- 9.Sokolov, V.V.: A new integrable case for Kirchhoff equations. Theoretical and Mathematical Physics. Nauka, Moscow 1(129), 31–37 (2001)Google Scholar
- 10.Borisov, A.V., Mamayev, I.S., Sokolov, V.V.: A new integrable case on so(4). Russian Math. Surveys 5(381), 614–615 (2001)Google Scholar
- 12.Irtegov, V.D.: Invariant Manifolds of Steady-State Motions and Their Stability. Nauka Publ., Novosibirsk (1985)Google Scholar