Sur de nouvelles séries pour le problème de masses critiques de Routh dans le problème restreint plan des trois corps |
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| Authors: | J Roels |
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| Institution: | 1. Institut de Mathématique, Université de Louvain, Belgique
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| Abstract: | The planar restricted 3-body problem, linearized in the neighborhood of Lagrangian equilibriaL 4 andL 5, has in general two distinct eigenvalues and their opposites. When they are pure imaginary and not multiples of each other, they generate two families of periodic solutions called long and short periodic families. This is essentially a consequence of the famous theorem of Liapunov (Siegel, 1956). We showed (Roels, 1971b) how to solve the problem when the eigenvalues are multiples of each other in building series with negative exponents instead of the integer expansions of Siegel (Roels and Lauterman, 1970). When the eigenvalues are equal, which is the case for the mass ratio of Routh, the problem was solved by Deprit and Henrard (1968) using formal series in ordinary unnormalized variables. That leads to very complicated series because of the use of variables that are not well adapted to the problem. The convergence of the series was proven by Meyer and Schmidt (1971). In this paper we solve the problem by using normalized variables. This brings us to build expansions with fractional exponents. So in summary, normalized variables generate integer series in the non-resonant cases, series with negative exponents in the case of resonancek≥3, and series with fractional exponents when the resonance is 1. |
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