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Contopoulos, G., 1963: Resonance cases and small divisors in a third integral of motion. I. Astron. J., 68, 763-779.
In this paper a general discussion of the resonance cases in an axially-symmetric potential field is presented, when the unperturbed frequencies in the radial and z direction have a rational ratio. The general form of the third integral is not valid in these cases because of the appearance of divisors of the form (m2P-n2Q), which become zero in the resonance cases. However, a new isolating integral of the unperturbed case is available, and this can be used to construct a third integral in the form of a power series and eliminate all secular terms. Three cases are distinguished, (α) m+n>4, (β) m+n=4, and (γ) m+n<4. In the first case the orbits are rather similar to those of the general irrational case. In the third case the orbits show a quite peculiar character, which, however, can be explained rather accurately by a first-order theory of the third integral. Numerical integrations were made for the cases P= 16Q, 4P=9Q, and P=4Q. The third integral, given in first- or second-order approximation, is rather well conserved. Case {3 and the cases of small divisors, when m2P-n2Q is near zero but not equal to zero, are discussed in Paper II.
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