A second-order solution of the Ideal Resonance Problem by Lie series |
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Authors: | Alan H Jupp |
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Institution: | 1. Department of Mathematics, North Carolina State University, Raleigh, N.C., USA
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Abstract: | A second-order libration solution of theIdeal Resonance Problem is construeted using a Lie-series perturbation technique. The Ideal Resonance Problem is characterized by the equations $$\begin{gathered} - F = B(x) + 2\mu ^2 A(x)sin^2 y, \hfill \\ \dot x = - Fy,\dot y = Fx, \hfill \\ \end{gathered} $$ together with the property thatB x vanishes for some value ofx. Explicit expressions forx andy are given in terms of the mean elements; and it is shown how the initial-value problem is solved. The solution is primarily intended for the libration region, but it is shown how, by means of a substitution device, the solution can be extended to the deep circulation regime. The method does not, however, admit a solution very close to the separatrix. Formulae for the mean value ofx and the period of libration are furnished. |
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