Resonance cases and small divisors in a third integral of motion. III
G. Contopoulos, Michael D. Moutsoulas · The Astronomical Journal · 1966
This paper discusses two cases where small divisors play an important role in the third integral. The Hamiltonian used is H = 21 (x2+ Y2+A x2+By2) - xy2 = h. In the first case the two unperturbed frequencies are nearly equal. If E is very small and we set B =A +KE2 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)i