Abstract
We solve a nonlinear orbital stability problem for a periodic motion of a homogeneous paraboloid of revolution over an immovable horizontal plane in a homogeneous gravity field. The plane is assumed to be absolutely smooth, and the body–plane collisions are assumed to be absolutely elastic. In the unperturbed motion, the symmetry axis of the body is vertical, and the body itself is in translational motion with periodic collisions with the plane.
The Poincare´ section surfacemethod is used to reduce the problemto studying the stability of a fixed point of an area-preserving mapping of the plane into itself. The stability and instability conditions are obtained for all admissible values of the problem parameters.
Similar content being viewed by others
References
I. G. Malkin, Theory of Stability of Motion (Nauka, Moscow, 1966) [in Russian].
P. Appel, Theoretical Mechanics, Vol. 2 (Fizmatgiz, Moscow, 1960) [in Russian].
A. P. Markeev, “Stability of Motion of a Rigid Body Colliding with a Horizontal Plane,” Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 5, 32–40 (1997) [Mech. Solids (Engl. Transl.) 32 (5), 27–34 (1997)].
V. Ph. Zhuravlev and D. M. Klimov, Applied Methods in Theory of Vibrations (Nauka, Moscow, 1988) [in Russian].
V. Ph. Zhuravlev and N. A. Fufaev, Mechanics of Systems with Unilateral Constraints (Nauka, Moscow, 1993) [in Russian].
A. P. Ivanov, Dynamics of Systems withMechanical Collisions (MEzhd. Progr. Obrazov., Moscow, 1997) [in Russian].
H. Poincaré, Selected Works, Vol. 2: New Methods of Celestial Mechanics (Nauka, Moscow, 1972) [in Russian].
A. P. Markeev, “Area-Preserving Mappings and Their Applications to the Dynamics of Systems with Collisions,” Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 2, 37–54 (1996) [Mech. Solids (Engl. Transl.) 31 (2), 32–47 (1996)].
A. P. Markeev, “On Stability of Fixed Points of Area-Preserving Mappings,” Nelin. Din. 11 (3), 503–545 (2015).
A. P. Markeev, Libration Points in Celestial Mechanics and Space Dynamics (Nauka, Moscow, 1978) [in Russian].
J. Moser, Lectures on Hamiltonian Systems (Amer. Math. Soc., Providence, 1968; Mir, Moscow, 1973).
C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics (Springer, New York, 1971; NITs “Regular and Chaotic Dynamics”, Izhevsk, 2001).
V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics (Editorial URSS, Moscow, 2002) [in Russian].
A. P. Markeev, “A Method for Analytically Representing Area-Preserving Mappings,” Prikl. Mat. Mekh. 78 (5), 611–624 (2014) [J. Appl. Math. Mech. (Engl. Transl.) 78 (5), 435–444 (2014)].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.P. Markeev, 2016, published in Izvestiya Akademii Nauk, Mekhanika Tverdogo Tela, 2016, No. 6, pp. 3–14.