1. In any case, what makes these periodic solutions so precious for us is that they are, so to speak, the only breach through which we can try to penetrate into a place so far deemed inaccessible“. H. Poincaré: Les Méthodes Nouvelles de la Mécanique Céleste (Gauthiers-Villars, Paris, 1892), Vol. 1, p. 82.
2. For C1 Hamiltonians, the point has been established by Pugh and Robinson. See C. Pugh, C. Robinson: The C1 closing lemma, including Hamiltonians. Erg. Theory and Dyn. Syst. 3, 261–313 (1977).
3. For a demonstration of this statement and for a further deepening of the subject, the reader is referred to C. L. Siegel, J. K. Moser: Lectures on Celestial Mechanics (Springer, 1971), Sect. 21, of which our exposition is an incomplete summary.
4. L. A. Pars: A Treatise on Analytical Dynamics (Heinemann, 1964) Sects. 30.7, 30.8.
5. V. Szebehely: Theory of Orbits (Academic Press, 1967) Sect. 8.6.