Is Jacobi theorem valid in the singly averaged restricted circular Three-Body-Problem?
DOI:
https://doi.org/10.21638/spbu01.2021.116Abstract
C. Jacobi found that in the General N-Body-Problem (including N = 3) for the Lagrangian stability of any solution necessary is the negativity of the total energy of the system. For the restricted three-body-problem, this statement is trivial, since a zero-mass body introduces zero contribution to the energy of the system. If we consider only the equations describing the movement of the zero mass point, then the energy integral disappears. However, if we average the equations over the longitudes of the main bodies, the energy integral appears again. Is the Jacobi theorem valid in this case? It turned out not. For arbutrary large values of total energy, there exist bounded periodic orbits. At the same time the negative energy is sufficient for the boundedness of an orbit in the configuration space.Keywords:
restricted circular Three-Body-Problem, Jacobi theorem on stability, averaging
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References
Литература
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References
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3. Kholshevnikov K.V., Titov V.B. Minimal velocity surface in the restricted circular Three Body-Problem. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 7 (65), iss. 4, 734–742 (2020). https://doi.org/10.21638/spbu01.2020.413 (In Russian) [Engl. transl.: Vestnik St. Petersb. Univ. Math. 53, iss. 4, 473–479 (2020)].
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Published
2021-05-29
How to Cite
Kholshevnikov, K. V. (2021). Is Jacobi theorem valid in the singly averaged restricted circular Three-Body-Problem?. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(1), 179–184. https://doi.org/10.21638/spbu01.2021.116
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Astronomy
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
