Abstract
The existence, stability, and branching of stationary motions of a tetrahedral body around a fixed point in the central Newtonian force field are studied. The case of an isosceles tetrahedron close to regular is considered. The connection of these properties of stationary motions with the properties of stationary motions of a regular tetrahedron is investigated. In celestial mechanics, the gravitational field of an irregularly shaped celestial body is often modeled by the gravitational field of some set of massive points. Representations in the form of a set of two or three point masses contain symmetries that are not inherent in real bodies. Approximations of gravitational fields with the help of fields of attraction of exactly four massive points seem to be the most suitable.
Similar content being viewed by others
REFERENCES
R. S. Sulikashvili, “Stationary motions of tetrahedron and octahedron in the central gravitational field,” in Problems of Stability and Motion Stabilization (Vych. Zentr AN SSSR, Moscow, 1987), pp. 57–66.
R. S. Sulikashvili, “On the stationary motions in a Newtonian field of force of a body that admits of regular polyhedron symmetry groups,” J. Appl. Math. Mech. 53 (4), 452–456 (1989). https://doi.org/10.1016/0021-8928(89)90051-8
A. A. Burov and R. S. Sulikashvili, “On the motion of a rigid body possessing a _nite group of symmetry,” Prépublication du C.E.R.M.A. Ecole Nationale des Ponts et Chaussées, No. 17 (1993).
A. A. Burov and E. A. Nikonova, “Rotation of isosceles tetrahedron in central newtonian force field: staude cone,” Moscow Univ. Mech. Bull. 76, 123–129 (2021). https://doi.org/10.3103/S0027133021050034
A. A. Burov and E. A. Nikonova, “Steady motions of a symmetric isosceles tetrahedron in a central force field,” Mech. Solids 56, 737–747 (2021). https://doi.org/10.3103/S0025654421050071
E. A. Nikonova, “On stationary motions of an isosceles tetrahedron with a fixed point in the central field of forces,” Prikl. Mat. Mekh. 86 (2), 153–168 (2022). https://doi.org/10.31857/S0032823522020096
A. V. Karapetyan and I. I. Naralenkova, ‘‘The bifurcation of the equilibria of mechanical systems with symmetrical potential,’’ J. Appl. Math. Mech. 62, 9–17 (1998). https://doi.org/10.1016/S0021-8928(98)00021-5
I. I. Naralenkova, ‘‘On the branching and stability of equilibrium positions of a rigid body in the Newtonian field,’’ in Problems of Stability and Motion Stabilization (Vych. Tsentr Ross. Akad. Nauk, Moscow, 1995), pp. 53–60.
Ye. V. Abrarova and A. V. Karapetyan, “Steady motions of a rigid body in a central gravitational field,” J. Appl. Math. Mech. 58 (5), 825–830 (1994). https://doi.org/10.1016/0021-8928(94)90007-8
Ye.V. Abrarova, “The stability of the steady motions of a rigid body in a central field,” J. Appl. Math. Mech. 59 (6), 903–910 (1995). https://doi.org/10.1016/0021-8928(95)00123-9
A. A. Burov and A. V. Karapetyan, “On the motion of cross-shaped bodies,” Mech. Solids 30 (6), 11–15 (1995).
Ye. V. Abrarova, “On the relative equilibria of a solid in a central gravitational field,” Problems of Stability and Motion Stabilization (Vych. Tsentr Ross. Akad. Nauk, Moscow, 1995), pp. 3–28.
Ye. V. Abrarova and A. V. Karapetyan, “Bifurcation and stability of the steady motions and relative equilibria of a rigid body in a central gravitational field,” J. Appl. Math. Mech. 60 (3), 369–380 (1996). https://doi.org/10.1016/S0021-8928(96)00047-0
A. A. Burov, A. D. Guerman, and R. S. Sulikashvili, “The orbital motion of a tetrahedral gyrostat,” J. Appl. Math. Mech. 74 (4), 425–435 (2010). https://doi.org/10.1016/j.jappmathmech.2010.09.008
A. A. Burov, A. D. Guerman, and R. S. Sulikashvili, “The steady motions of gyrostats with equal moments of inertia in a central force field,” J. Appl. Math. Mech. 75 (5), 517–521 (2011). https://doi.org/10.1016/j.jappmathmech.2011.11.005
A. A. Burov, A. D. Guerman, and R. S. Sulikashvili, “Dynamics of a tetrahedral satellite-gyrostat,” AIP Conf. Proc. 1281, 465–468 (2010). https://doi.org/10.1063/1.3498509
A. A. Burov, A. D. Guerman, E. A. Nikonova, V. I. Nikonov, “Approximation for attraction field of irregular celestial bodies using four massive points,” Acta Astronaut. 157, 225–232 (2019). https://doi.org/10.1016/j.actaastro.2018.11.030
H. Yang, Sh. Li, and J. Sun, “A fast Chebyshev polynomial method for calculating asteroid gravitational fields using space partitioning and cosine sampling,” Adv. Space Res. 65 (4), 1105–1124 (2020). https://doi.org/10.1016/j.asr.2019.11.001
V. N. Rubanovsky and V. A. Samsonov, Stability of Stationary Motions in Examples and Problems (Nauka, Moscow, 1988) [in Russian].
E. J. Routh, Treatise on the Stability of a Given State of Motion (Cambridge Uni. press, Cambridge, 1877).
E. J. Routh, The Advanced Part of a Treatise on the Dynamics of a System of Rigid Rodies (MacMillan, London, 1884).
A. V. Karapetyan, Stability of Stationary Movements (Editorial URSS, Moscow, 1998) [in Russian].
I. F. Sharygin, Problems in Geometry. Stereometry (Nauka, Moscow, 1984) [in Russian].
M. A. Vashkoviak, “On the stability of circular “asteroid” orbits in an N-planetary system,” Celest. Mech. 13 (3), 313–324 (1976).
A. A. Burov and V. I. Nikonov, “Stability and branching of stationary rotations in a planar problem of motion of mutually gravitating triangle and material point,” Rus. J. Nonlin. Dyn. 12 (2), 179–196 (2016). https://doi.org/10.20537/nd1602002
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Nikonova, E.A. Stationary Motions of a Close to Regular Isosceles Tetrahedron with a Fixed Point in the Central Newtonian Force Field. Mech. Solids 57, 1059–1067 (2022). https://doi.org/10.3103/S0025654422050211
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0025654422050211