×

Separatrix crossing in rotation of a body with changing geometry of masses. (English) Zbl 1446.70014

Summary: We consider free rotation of a body whose parts move slowly with respect to each other under the action of internal forces. This problem can be considered as a perturbation of the Euler-Poinsot problem. The dynamics has an approximate conservation law – an adiabatic invariant. This allows us to describe the evolution of rotation in the adiabatic approximation. The evolution leads to an overturn in the rotation of the body: the vector of angular velocity crosses the separatrix of the Euler-Poinsot problem. This crossing leads to a quasirandom scattering in body’s dynamics. We obtain formulas for probabilities of capture into different domains in the phase space at separatrix crossings.

MSC:

70E20 Perturbation methods for rigid body dynamics
70E55 Dynamics of multibody systems
70H11 Adiabatic invariants for problems in Hamiltonian and Lagrangian mechanics
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
70K65 Averaging of perturbations for nonlinear problems in mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] V. I. Arnol’d, {\it Small denominators and problems of stability of motion in classical and celestial mechanics}, Russian Math. Surveys, 18 (1963), pp. 85-191. · Zbl 0135.42701
[2] V. I. Arnol’d, V. V. Kozlov, and A. I. Neishtadt, {\it Mathematical Aspects of Classical and Celestial Mechanics}, 3rd ed., Springer-Verlag, New York, 2006. · Zbl 1105.70002
[3] R. Bellman, {\it Introduction to Matrix Analysis}, McGraw-Hill Book, New York, 1960. · Zbl 0124.01001
[4] A. V. Borisov and I. S. Mamaev, {\it Adiabatic invariants, diffusion and acceleration in rigid body dynamics}, Regul. Chaotic Dyn., 21 (2016), pp. 232-248. · Zbl 1377.70021
[5] F. L. Chernousko, L. D. Akulenko, and D. D. Leshchenko, {\it Evolution of Motions of a Rigid Body about Its Center of Mass}, Springer International Publishing, New York, 2017. · Zbl 1371.70002
[6] A. Deprit, {\it Free rotation of a rigid body studied in the phase plane}, Am. J. Phys., 35 (1967), pp. 424-428.
[7] P. Goldreich and S. Peale, {\it Spin-orbit coupling in the solar system}, Astron. J., 71 (1966), pp. 425-438.
[8] P. Goldreich and A. Toomre, {\it Some remarks on polar wandering}, J. Geophys. Res., 74 (1969), pp. 2555-2567.
[9] M. Lara and S. Ferrer, {\it Integration of the Euler-Poinsot Problem in New Variables}, preprint, [nlin.SI], 2010.
[10] I. M. Lifshits, A. A. Slutskin, and V. M Nabutovskii, {\it The scattering of charged quasi-particles from singularities in p-space}, Dokl. Phys., 6(1961), pp. 238-240.
[11] K. W. Lips and V. J. Modi, {\it Three-axis attitude dynamics during asymmetric deployment of flexible appendages}, Acta Astronaut., 8 (1981), pp. 575-590.
[12] I. Matsuyama, F. Nimmo, and J. X. Mitrovica, {\it Planetary reorientation}, Annu. Rev. Earth Planet. Sci., 42 (2014), pp. 605-634.
[13] W. Munk and G. J. F. MacDonald, {\it The Rotation of the Earth}, Cambridge University Press, Cambridge, MA, 1960. · Zbl 1159.86300
[14] A. I. Neishtadt, {\it Passage through a separatrix in a resonance problem with a slowly-varying parameter}, J. Appl. Math. Mech., 39 (1975), pp. 594-605. · Zbl 0356.70020
[15] A. I. Neishtadt, {\it Averaging method for systems with separatrix crossing}, Nonlinearity, 30 (2017), pp. 2871-2917. · Zbl 1381.37073
[16] E. Routh, {\it Dynamics of Systems of Rigid Bodies}, Part II, Macmillan, New York, 1905. · JFM 17.0315.02
[17] Y. A. Sadov, {\it The action-angle variables in the Euler-Poinsot problem}, J. Appl. Math. Mech., 34 (1970), pp. 922-925. · Zbl 0224.70002
[18] K. Tsuchiya, {\it Dynamics of a spacecraft during extension of flexible appendage}, J. Guid. Control Dyn., 6 (1983), pp. 100-103. · Zbl 0521.70030
[19] A. E. Zakrzhevskii, J. Matarazzo, and V. S. Khoroshilov, {\it Dynamics of a system of bodies with program-variable configuration}, Internat. Appl. Mech., 40 (2004), pp. 345-350.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.