Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 049, 20 pages      math.CA/0605204      https://doi.org/10.3842/SIGMA.2006.049

On One Approach to Investigation of Mechanical Systems

Valentin D. Irtegov and Tatyana N. Titorenko
Institute for Systems Dynamics and Control Theory, SB RAS, Irkutsk, Russia

Received November 18, 2005, in final form April 11, 2006; Published online May 08, 2006

Abstract
The paper presents some results of qualitative analysis of Kirchhoff's differential equations describing motion of a rigid body in ideal fluid in Sokolov's case. The research methods are based on Lyapunov's classical results. Methods of computer algebra implemented in the computer algebra system (CAS) "Mathematica" were also used. Combination of these methods allowed us to obtain rather detailed information on qualitative properties for some classes of solutions of the equations.

Key words: rigid body mechanics; completely integrable systems; qualitative analysis; invariant manifolds; stability; bifurcations; computer algebra.

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References

  1. Borisov A.V., Mamaev I.S., Sokolov V.V., A new integrable case on so(4), Dokl. Akad. Nauk, 2001, V.381, N 5, 614-615 (English transl.: Dokl. Phys., 2001, V.46, N 12, 888-889).
  2. Cox D., Little J., O'Shea D., Ideals, varieties, and algorithms, New York, Springer-Verlag, 1997.
  3. Harlamov P.V., On motion in a fluid of a body bounded by a multiply connected surface, Prikl. Mekh. Tekhn. Fiz., 1963, N 4, 17-29 (in Russian).
  4. Kirchhoff G., Vorlesungen über Mathematische Physik. Mechanik, Leipzig, B. Teubner, Bd. 1, 1897.
  5. Kovalev Yu.M., On the stability of steady helical motions in a fluid of a body bounded by a multiply connected surface, J. Appl. Math. Mech., 1968, V.32, N 2, 272-275.
  6. Kozlov V.V., Onishchenko D.A., The motion in a perfect fluid of a body containing a moving point mass, J. Appl. Math. Mech., 2003, V.67, N 4, 553-564.
  7. Lamb H., Hydrodynamics, New York, Dover Publ., 1945.
  8. Lyapunov A.M., On permanent helical motions of a rigid body in fluid, Moscow-Leningrad, USSR Acad. Sci., Collected Works, Vol. 1, 1954 (in Russian).
  9. Lyapunov A.M., Stability of motion, New York, Academic Press, 1966.
  10. Oparina E.I., Troshkin O.V., Stability of Kolmogorov flow in a channel with rigid walls, Dokl. Akad. Nauk, 2004, V.398, N 4, 487-491 (English transl.: Dokl. Phys., 2004, V.49, N 10, 583-587).
  11. Sokolov V.V., A new integrable case for the Kirchhoff equation, Teoret. Mat. Fiz., 2001, V.129, N 1, 31-37 (English transl.: Theoret. and Math. Phys., 2001, V.129, N 1, 1335-1340).
  12. Rumyantsev V.V., A comparison of three methods of constructing Lyapunov functions, J. Appl. Math. Mech., 1995, V.59, N 6, 873-877.
  13. Ryabov P.E., Bifurcations of first integrals in the Sokolov case, Teoret. Mat. Fiz., 2003, V.134, N 2, 207-226 (English transl.: Theoret. and Math. Phys., 2003, V.34, N 2, 181-197).

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