Abstract
Numerical evidence is presented which indicates that, although the third integral is tangent to the Hamiltonian (energy integral) along some periodic orbits (as has been shown by Goudas), it is not tangent to it along non-periodic orbits; therefore it is not a function of the Hamiltonian. The set of periodic orbits is probably dense in general, but a given form of the third integral is valid in the neighbourhood of a limited number of them; no form of the third integral is valid for all periodic orbits, except in integrable cases.
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Contopoulos, G., Barbanis, B. Is the third integral a function of the Hamiltonian?. Astrophys Space Sci 2, 134–145 (1968). https://doi.org/10.1007/BF00651501
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DOI: https://doi.org/10.1007/BF00651501