Motion of a variable body in a time-dependent force field

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

The problem of translational-rotational motion of a variable body is considered under the assumption that the inertial properties of the body, as well as the external forces acting on it and the moments of forces clearly depend on time. The conditions under which the equations of motion are reduced to classical equations describing the motion of a solid body in a force field independent of time are indicated. There are cases when the equations of motion are reduced to completely integrable ones. The elements of the discussion of the 1920-1930 on the description of the motion of a material point of variable mass in a time-dependent field of attraction are reproduced.

About the authors

A. A. Burov

Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences

Author for correspondence.
Email: jtm@narod.ru
Russian Federation, Moscow

V. I. Nikonov

Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences

Email: nikon_v@list.ru
Russian Federation, Moscow

References

  1. Borisov, A.V. and Mamaev, I.S. Rigid Body Dynamics. Hamiltonian Methods, Integrability, Chaos. Institute of computer Science, Moscow, 2005. 576 p.
  2. Levi-Civita T. Sul mote di un corpo di massa variabile // Rendiconti delle Sedute della Reale Accademia dei Lincei. 1928. V. 8. P. 329–333.
  3. Levi-Civita T. Ancora sul moto di un corpo di massa variabile // Rendiconti delle Sedute della Reale Accademia dei Lincei. 1930. V. 11. P. 626 – 632.
  4. Doubochine G. Mouvement d’un point matériel sous l’action d’une force qui déepend du temps // Russian Astronomical Journal. 1925. V. 2. № 4. P. 5–11.
  5. Doubochine G. Mouvement d’un point matériel sous l’action d’une force qui déepend du temps. II // Russian Astronomical Journal. 1927. V. 4. № 2. P. 123–141.
  6. Doubochine G.N. On the motion of a material point under the action of a time-dependent force. II // Russian Astronomical Journal. 1927. V. 4. № 2. P. 141–142.
  7. Doubochine G.N. On the motion of a material point under the action of a time-dependent force. III. Investigation of a particular case // Russian Astronomical Journal. 1928. V. 5. № 2–3. P. 138–151.
  8. Doubochine G. Mouvement d’un point matériel sous l’action d’une force qui déepend du temps. IV. Une méthode nouvelle pour la resolution du problème // Russian Astronomical Journal. 1929. V. 6. № 2. P. 162–179.
  9. Stepanoff W. Sur la forme de trajectoires d’un point matériel dans le cas de l’attraction Newtonienne d’une masse variable // Russian Astronomical Journal. 1930. V. 7. № 2. P. 73–80.
  10. Doubochine G. Sur la forme de trajectoires dans le problème des deux corps de masse variables. // Astronomical Journal. 1930. V. 7. № 3–4. P. 153–172.
  11. Mestschersky I. The dynamics of a point of variable mass. St. Petersburg: Type. Imp. AN. 1897. 160 p.
  12. Gyldén H. Die Bahnbewegungen in einem Systeme von zwei Körpern in dem Falle, dass die Massen Veränderungen unterworfen sind // Astronomische Nachrichten. 1884. V. 109. № 1. P. 2593–2594 (1–4).
  13. Mestschersky J. Ein Specialfall des Gyldén’schen Problems. (A. N.2593) // Astronomische Nachrichten. 1893. V. 132. № 9. P. 3153 (129–130).
  14. Mestschersky J. Über die Integration der Bewegungsgleichungen im Probleme zweier Körper von veränderlicher Masse // Astronomische Nachrichten. 1902. V. 159. № 15. P. 3807 (231–242).
  15. Berkovich L.M. The Gilden–Meshchersky problem and the laws of mass change // Dokl. USSR Academy of Sciences. 1980. V. 250. № 5. С. 1088 – 1091.
  16. Bekov A.A. Dynamics of double nonstationary gravitational systems. Almaty: Gylym. 2013. 170 с.
  17. Ong J.J., O’Reilly O.M. On the equations of motion for rigid bodies with surface growth // Int. J. Eng. Science. 2004. V. 42. № 19–20. P. 2159–2174. https://doi.org/10.1016/j.ijengsci.2004.07.010.
  18. Irschik H., Humer A. A rational treatment of the relations of balance for mechanical systems with a time-variable mass and other non-classical supplies // Dynamics of Mechanical Systems with Variable Mass. International Centre for Mechanical Sciences Courses and Lectures. V. 557. 2014. P. 1–50. https://doi.org/10.1007/978-3-7091-1809-2_1.
  19. Cveticanin L. Dynamics of Machines with Variable Mass. 1st edition. London: Routledge. 1998. 300 p. https://doi.org/10.1201/9780203759066
  20. Zeiliger D.N. Theory of motion of a similarly changeable body. Kazan: type. Kazan Imperial University. 1892. 105 p.
  21. Chetaev N.G. On the equations of motion of a similarly variable body // Scientific notes of the Kazan University. 1954. Vol. 114. P. 5–7.
  22. Chetaev N.G. Theoretical mechanics. M.: Nauka. 1987. 368 p.
  23. Sławianowski J.J. The mechanics of the homogeneously-deformable body. Dynamical models with high symmetries // ZAMM. 1982. V. 62. № 6. P. 229–240. https://doi.org/10.1002/zamm.19820620604.
  24. Sławianowski J.J. Affinely rigid body and Hamiltonian systems on GL(n,R) // Reports on Mathematical Physics. 1988. V. 26. № 1. P. 73–119. https://doi.org/10.1016/0034-4877(88)90006-7.
  25. Sławianowski J.J., Kovalchuk V., Golubowska B. et al. Mechanics of affine bodies. Towards affine dynamical symmetry // J. Math. Anal. Appl. 2017. V. 446. № 1. https://doi.org/10.1016/j.jmaa.2016.08.042.
  26. Burov A.A., Chevallier D.P. Dynamics of affinely deformable bodies from the standpoint of theoretical mechanics and differential geometry // Reports on Math. Phys. 2008. V. 62. № 3. P. 283–321. https://doi.org/10.1016/S0034-4877(09)00003-2
  27. Burov A., Guerman A., Kosenko I. Satellite with periodical mass redistribution: relative equilibria and their stability // Celest. Mech. Dyn. Astr. 2019. V. 131. P. 1 (1–12). https://doi.org/10.1007/s10569-018-9874-0

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2024 Russian Academy of Sciences