Geodesic
Cases of integrability corresponding to the pendulum motion in three-dimensional space
Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 3-4 (2016), pp. 75-97 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this actitity, we systemize some results on the study of the equations of spatial motion of dynamically symmetric fixed rigid bodies-pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of a spatial motion of a free rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint. The obtained results are systematized and served in the invariant form. We also show the nontrivial topological and mechanical analogies.
Keywords: rigid body, resisting medium, dynamical system, three-dimensional phase pattern, case of integrability. 
@article{VSGU_2016_3-4_a6,
     author = {M. V. Shamolin},
     title = {Cases of integrability corresponding to the pendulum motion in three-dimensional space},
     journal = {Vestnik Samarskogo universiteta. Estestvennonau\v{c}na\^a seri\^a},
     pages = {75--97},
     year = {2016},
     number = {3-4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGU_2016_3-4_a6/}
}
TY  - JOUR
AU  - M. V. Shamolin
TI  - Cases of integrability corresponding to the pendulum motion in three-dimensional space
JO  - Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ
PY  - 2016
SP  - 75
EP  - 97
IS  - 3-4
UR  - http://geodesic.mathdoc.fr/item/VSGU_2016_3-4_a6/
LA  - ru
ID  - VSGU_2016_3-4_a6
ER  - 
%0 Journal Article
%A M. V. Shamolin
%T Cases of integrability corresponding to the pendulum motion in three-dimensional space
%J Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ
%D 2016
%P 75-97
%N 3-4
%U http://geodesic.mathdoc.fr/item/VSGU_2016_3-4_a6/
%G ru
%F VSGU_2016_3-4_a6
M. V. Shamolin. Cases of integrability corresponding to the pendulum motion in three-dimensional space. Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 3-4 (2016), pp. 75-97. http://geodesic.mathdoc.fr/item/VSGU_2016_3-4_a6/

[1] Shamolin M. V., “Cases of integrability corresponding to the pendulum motion on the plane”, Vestnik of Samara State University. Natural Sciences Series, 2015, no. 10(132), 91–113 (in Russ.)

[2] Shamolin M. V., “New integrable cases and families of portraits in the plane and spatial dynamics of a rigid body interacting with a medium”, Journal of Mathematical Sciences, 114:1 (2003), 919–975 | DOI | MR | Zbl

[3] Shamolin M. V., “Variety of cases of integrability in dynamics of lower-, and multidimensional body in nonconservative field”, Dynamical Systems, Contemporary Mathematics and its Applications, 125, 2013, 5–254. (in Russ.)

[4] Pokhodnya N. V., Shamolin M. V., “New case of integrability in dynamics of multi-dimensional body”, Vestnik of Samara State University. Natural Sciences Series, 2013, no. 9/1(110), 35–41 (in Russ.)

[5] Shamolin M. V., “Variety of types of phase portraits in dynamics of a rigid body interacting with a resisting medium”, Physics Doklady, 349:2 (1996), 193–197 (in Russ.) | Zbl

[6] Shamolin M. V., “Dynamical Systems With Variable Dissipation: Approaches, Methods, and Applications”, Journal of Mathematical Sciences, 14:3 (2008), 3–237 (in Russ.) | Zbl

[7] Arnold V. I., Kozlov V. V., Neyshtadt A. I., Mathematical aspect in classical and celestial mechanics, VINITI, M., 1985, 304 pp. (in Russ.)

[8] Trofimov V. V., “Symplectic structures on symmetruc spaces automorphysm groups”, Moscow University Mathematics Bulletin, 1984, no. 6, 31–33 (in Russ.)

[9] Trofimov V. V., Shamolin M. V., “Geometrical and dynamical invariants of integrable Hamiltonian and dissipative systems”, Journal of Mathematical Sciences, 16:4 (2010), 3–229 (in Russ.)

[10] Shamolin M. V., Methods of analysis of various dissipation dynamical systems in dynamics of a rigid body, Ekzamen, M., 2007, 352 pp. (in Russ.)

[11] Shamolin M. V., “Classes of variable dissipation systems with nonzero mean in the dynamics of a rigid body”, Journal of Mathematical Sciences, 122:1 (2004), 2841–2915 | DOI | MR | Zbl

[12] Shamolin M. V., “Some model problems of dynamics for a rigid body interacting with a medium”, International Applied Mechanics, 43:10 (2007), 49–67

[13] Shamolin M. V., “New integrable cases in dynamics of a medium-interacting body with allowance for dependence of resistance force moment on angular velocity”, Journal of Applied Mathematics and Mechanics, 72:2 (2008), 273–287 | Zbl

[14] Shamolin M. V., “Integrability of some classes of dynamic systems in terms of elementary functions”, Moscow University Mechanics Bulletin, 2008, no. 3, 43–49

[15] Shamolin M. V., “Stability of a rigid body translating in a resisting medium”, International Applied Mechanics, 45:6 (2009), 125–140 | Zbl