Abstract

view Abstract Citations (87) References (6) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Formal solution in the problem of small divisors Garfinkel, Boris Abstract Small divisors in mechanics arise from resonance, i.e., the near-commensurability of two of the natural frequencies of the motion. Under very general assumptions, the problem can be reduced to a canonical form of one degree of freedom, with the Hamiltonian F defined by F=Ao(y)+Ao(y) cos2z. Here z is the coordinate, y the momentum, and the coefficients A0 and A0 satisfy the relation A1/A0=O(k), where k is a small parameter. Resonance is associated with the vanishing of the derivative A 0' for some value Y=Y* In the neighborhood of Y*, the resonance problem is solved to O(k-") by the Bohlin-von Zeipel technique, and the solution is expressed in terms of elliptic functions. It is shown that the resonance solution reduces to the classical solution with a singularity at Y=Y*, when the divisor * is sufficiently large. Applications to celestial mechanics are illustrated by the problem of critical inclination and by the 24-h satellite in the artificial satellite theory. Publication: The Astronomical Journal Pub Date: October 1966 DOI: 10.1086/110171 Bibcode: 1966AJ.....71..657G full text sources ADS |

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