×

Qualitative analysis of the phase flow of a Manev system in a rotating reference frame. (English) Zbl 1175.70011

Summary: The rotating two-body Manev problem is defined by means of the Hamiltonian function \(\mathcal H = (p_r^2+p_{\theta}^2/r^2)/2 +\alpha p_\theta - 1/r +\beta / r^2\) with \((\alpha , \beta )\in \mathbb R^{+}\times \mathbb R\) being two structural parameters. Using the Liouville-Arnold theorem and a particular analysis of the momentum map in its critical points, we obtain a complete topological classification of the different invariant sets of the phase flow of this problem. This analysis, in some aspects very computational, is made with the help of a standard commercial mathematical package.

MSC:

70F07 Three-body problems
70K42 Equilibria and periodic trajectories for nonlinear problems in mechanics
70H14 Stability problems for problems in Hamiltonian and Lagrangian mechanics
53D17 Poisson manifolds; Poisson groupoids and algebroids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abraham R., Foundations and Mechanics (1978)
[2] Arnold V. I., Mathematical Methods of Classical Mechanics (1978) · Zbl 0386.70001
[3] Arnold V. I., Dynamical Systems III (Encyclopaedia of Mathematical Sciences) (1978)
[4] Arnold V. I., Mathematical Aspects of Classical and Celestial Mechanics (1997) · Zbl 0885.70001
[5] Balsas, M. B., Jiménez, E. S. and Vera, J. A.The Kepler Problem in Rotatin Reference Frames: Topological Study of the Phase Flow. AIP Conference Proceedings. Vol. 963, pp.1146–1449.
[6] DOI: 10.1023/B:JOSS.0000015177.60491.3c · Zbl 0917.45007 · doi:10.1023/B:JOSS.0000015177.60491.3c
[7] DOI: 10.1088/0305-4470/34/9/309 · Zbl 1006.70010 · doi:10.1088/0305-4470/34/9/309
[8] Manev G., Comptes Rendues 178 pp 2159– (1924)
[9] DOI: 10.1007/BF02980633 · doi:10.1007/BF02980633
[10] Manev G., Comptes Rendues 190 pp 963– (1930)
[11] Manev G., Comptes Rendues 190 pp 1374– (1930)
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.