Abstract
The Ideal Resonance Problem, defined by the HamiltonianF=B(y)+2εA (y) sin2 x, ε≪1, has been solved in Garfinkelet al. (1971). There the solution has beenregularized by means of a special functionφ j , introduced into the new HamiltonianF′, under the tacit assumption thatA anB¨' are of order unity.
This assumption, violated in some applications of the theory, is replaced here by the weaker assumption ofnormality, which admits zeros ofA andB′ inshallow resonance. It is shown here that these zeros generate singularities, which can be suppressed ifφ j is suitably redefined.
With the modifiedφ j , and with the assumption of normality, the solution is regularized for all values ofB′, B¨', andA. As in the previous paper, the solution isglobal, including asymptotically the classical limit withB′ as a divisor of O(1).
A regularized first-order aloorithm is constructed here as an illustration and a check.
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Garfinkel, B. Regularization in the Ideal Resonance Problem. Celestial Mechanics 5, 189–203 (1972). https://doi.org/10.1007/BF01229521
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DOI: https://doi.org/10.1007/BF01229521