Non-Continuation of Integrals of the Rotating Two-body Problem in Hill's Lunar Problem |
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| Authors: | Fabian Josef Winterberg Efi Meletlidou |
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| Institution: | (1) Swiss Federal Institute of Technology ETH, CH-8092 Zürich, Switzerland;(2) Department of Physics, University of Thessaloniki, GR-54006 Thessaloniki, Greece |
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| Abstract: | We consider Hill's lunar problem as a perturbation of the integrable two-body problem. For this we avoid the usual normalization in which the angular velocity  of the rotating frame of reference is put equal to unity and consider as the perturbation parameter. We first express the Hamiltonian H of Hill's lunar problem in the Delaunay variables. More precisely we deduce the expressions of H along the orbits of the two-body problem. Afterwards with the help of the conserved quantities of the planar two-body problem (energy, angular momentum and Laplace–Runge–Lenz vector) we prove that Hill's lunar problem does not possess a second integral of motion, independent of H, in the sense that there exist no analytic continuation of integrals, which are linear functions of in the rotating two-body problem. In connection with the proof of this main result we give a further restrictive statement to the nonintegrability of Hill's lunar problem. |
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| Keywords: | Hill's lunar problem Delaunay variables nonintegrability Laplace– Runge– Lenz vector |
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