Mathematics
Scientific paper
Apr 1976
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1976a%26a....48..275s&link_type=abstract
Astronomy and Astrophysics, vol. 48, no. 2, Apr. 1976, p. 275-280. In French.
Mathematics
Canonical Forms, Equations Of Motion, Existence Theorems, Hamiltonian Functions, Partial Differential Equations, Three Body Problem, Celestial Mechanics, Entire Functions, Kepler Laws, Periodic Functions, Perturbation Theory, Trajectory Analysis, Transformations (Mathematics)
Scientific paper
With the aid of a canonical formulation of the differential equations of the trajectories of the Jacobi constant in the planar restricted three-body problem, the author obtains an indirect demonstration of the existence of the second integral. The analysis is based on a Hamiltonian of one degree of freedom which is periodic in the independent variable (the polar angle) and which is then written in terms of a small parameter taken to be the inverse of the distance between the primaries. This manner of writing the Hamiltonian permits the application of the canonical averaging method, a procedure based on the search for a periodic canonical transformation such that the new Hamiltonian does not depend explicitly on the independent variable. The existence of certain classes of periodic solutions leads to the conclusion that the second integral, for a given Jacobi constant, exists on a discontinuous set of closed curves in the two-dimensional phase space of the representation.
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