Invariant Sets of Mechanical Systems
There are discussed problems of the existence and stability of invariant sets of mechanical systems (in particular, steady motions and relative equilibria).
The existence problem of stable motions (zero-dimensional invariant sets) firstly was investigated in the . Really, the famous Routh theory – gives stability conditions of steady motions of conservative mechanical systems with first integrals as well as construction method of such steady motions. This method was spreaded for the construction problem of steady motions not only stable ,  and for dissipative system with first integrals –.
Moreover the Routh theory was modified to the existence and stability problems of invariant sets of dynamical systems with an ungrowing function and first integrals ,  (in particular, of conservative and dissipative mechanical systems with symmetry –).
KeywordsRelative Equilibrium Steady Motion Dissipative Force Local Strict Minimum Total Mechanical Energy
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- 1.Routh, E.J.: A treatise on the Stability of a Given State of Motion, MacMillan and Co., London 1877.Google Scholar
- 2.Routh, E.J.: The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies, MacMillan and Co., London 1884.Google Scholar
- 4.Lyapunov, A.M.: On Constant Screw Motions of a Rigid Body in a Liquid, Izd. Khar’kov Mat. Obshch, Khar’kov 1888 (in Russian).Google Scholar
- 5.Lyapunov, A.M.: The General Problem of the Stability of Motion, Izd. Khar’kov Mat. Obshch, Khar’kov 1888 (in Russian).Google Scholar
- 8.Salvadori, L.: Un osservazione su di un criteria di stabilita dei Routh, Rend. Accad. Sci. Fis. Math. Napoli (IV), 20 (1953), 267–272.Google Scholar
- 9.Pozharitskii, G.K.: On the construction of the Lyapunov function from integrals of equations of perturbed motion, Prikl. Mat. Mekh., 22 (1958), 145–154 (in Russian)Google Scholar
- 10.Rumyantsev, V.V.: On the stability of permanent rotations of mechanical systems, Izv. AN SSSR, OTN, Mekh. Mash., 6 (1962), 113–121 (in Russian).Google Scholar
- 11.Rumyantsev, V.V.: On the stability of steady motions, Prikl. Mat. Mekh. 30 (1966), 922–923 (in Russian).Google Scholar
- 14.Karapetyan, A.V.: The Routh Theorem and its Extensoins, in: Colloq. Math. Soc. Janos Bolyai, 53: Qualitative Theory of Differential Equations, North Holland, Amsterdam and New York 1990, 271–290.Google Scholar
- 15.Karapetyan, A.V. and Rumyantsev, V.V.: Stability of Conservative and Dissipative Systems, in: Appl. Mech. Soviet Reviews, 1: Stability and Anal. Mech., Hemisphere, New York 1990, 3–144.Google Scholar
- 17.Abraham, R. and Marsden J.: Foundations of Mechanics, Benjamin, New York and Amsterdam 1978.Google Scholar
- 18.Karapetyan A.V.: Invariant Sets of Mechanical Systems with Symmetry, in: Vychisl. Mat and Inform., VTs RAN, Moscow 1996, 74–86 (in Russian).Google Scholar
- 20.Tomson, W. and Tait, P.: Treatise on Natural Phylosophy, Cambridge Univ. Press, Cambridge (1879).Google Scholar
- 21.Chetaev, N.G.: The Stability of Motion, Pergamon Press, London (1961)Google Scholar
- 22.Krasovskii, N.N.: Problems of the Theory of Stability of Motion, Stanford Univ. Press, Stanford (1963).Google Scholar
- 24.Stepanov, S.Ja.: On the Relationship of the Stability Conditions for Three Different Regimes of Cyclic Motions in a System, in: Probi. Anal. Mekh., Teor. Ust. i Upr., KAI, Kazan’ (2) 1976, 303–308 (in Russian).Google Scholar
- 26.Karapetyan, A.V. and Stepanov, S.Ja.: On the relationship of stability conditions of steady motions of a free system and positions of relative equilibrium of a restricted system, Sb. nauchn method. statei po teor. mekh., 20 (1990), 31–37 (in Russian)Google Scholar
- 28.Kozlov, V.V.: On the degree of instability, Prikl. Math. Mekh., 57 (1993), 14–19 (in Russian).Google Scholar