Open AccessIs Jacobi theorem valid in the singly averaged restricted circular Three-Body-Problem?
Author(s) -
К. В. Холшевников
Publication year - 2021
Publication title -
vestnik sankt-peterburgskogo universiteta. matematika. mehanika. astronomiâ/vestnik sankt-peterburgskogo universiteta. seriâ 1, matematika, mehanika, astronomiâ
Language(s) - English
Resource type - Journals
eISSN - 2587-5884
pISSN - 1025-3106
DOI - 10.21638/spbu01.2021.116
Subject(s) - mathematics , three body problem , zero (linguistics) , bounded function , energy (signal processing) , mathematical analysis , zero point energy , orbit (dynamics) , space (punctuation) , physics , classical mechanics , quantum mechanics , linguistics , philosophy , statistics , engineering , aerospace engineering
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.