Publication Abstracts

Contopoulos and Moutsoulas 1966

Contopoulos, G., and M. Moutsoulas, 1966: Resonance cases and small divisors in a third integral of motion. III. Astron. J., 71, 687-698, doi:10.1086/110173.

This paper discusses two cases where small divisors play an important role in the third integral. The Hamiltonian used is H = (1/2)(X2+Y2+Ax2+By2)-εxy2 = h h. In the first case the two unperturbed frequencies are nearly equal. If ε is very small and we set B = A+Kε2 we find resonance phenomena when K is in the range ( -5h/3A2, 10h/3A2). For larger or smaller values of K all the orbits are boxes. This range is divided into four parts by the values K = -2h/3A2, K = 5h/6A2, and K = 7h/3A2. The forms of the invariant curves are different in the four intervals. The corresponding orbits are either box type, or similar to the orbits of the resonance case A =B, except for the D-type orbits, which appear only in the third and fourth intervals.

In the second case one unperturbed frequency is almost the double of the other. If we set 4B = A+ek we find resonance phenomena when -4(2h/A)1/2 < k < 4(2h/A)1/2. The boundaries of the orbits are approximately arcs of three parabolas, as in the resonance case.

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BibTeX Citation

@article{co07200r,
  author={Contopoulos, G. and Moutsoulas, M.},
  title={Resonance cases and small divisors in a third integral of motion. III},
  year={1966},
  journal={Astron. J.},
  volume={71},
  pages={687--698},
  doi={10.1086/110173},
}

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RIS Citation

TY  - JOUR
ID  - co07200r
AU  - Contopoulos, G.
AU  - Moutsoulas, M.
PY  - 1966
TI  - Resonance cases and small divisors in a third integral of motion. III
JA  - Astron. J.
VL  - 71
SP  - 687
EP  - 698
DO  - 10.1086/110173
ER  -

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