ALBERT

Description / Table of Contents: Summary The conservative differential equations (either independant of the time or periodical) are very frequent, for example those given by a canonical system the Hamiltonian of which is sufficiently regular and either independent of the time or periodical. The topological study of these equations leads to 4 main types of trajectories: (1) The ‘open trajectories’; coming from infinity and going to infinity. (2) The ‘limited trajectories’; they always come again into any neighbourhood of any past position. (3) The ‘oscillating trajectories’; they also always come again into any neighbourhood of any past position, but they also go to infinity. (4) The ‘abnormal trajectories’; infinitely rare and much more complicated, such as the trajectories which are asymptotic to an unstable periodic trajectory. All of these types may be encountered in the three-body problem of Celestial Mechanics. In particular the ‘oscillating trajectories’ which theoretically lead to infinitesimal close passages (and then to infinitely large velocities) lead practically to a collision. Hence the collision of stars after, for instance, the encounter of two binary systems, are 103 to 105 times more frequent than the simple kinematic point of view may lead one to think. The study utilizes methods of modern mathematics, especially set theory and the theory of measure. First the time is eliminated and the study of a space transformation (mapping) takes the place of the study of the trajectories. The main property of this mapping is to preserve the measures. Then the intersection of a trajectory and a measurable set is studied. This leads to three main types of intersections which lead to the first three types of trajectories mentioned above. The abnormal trajectories are studied separately. The application of the study to the trajectories of the three-body problem leads to the following modes: Let us callR the maximal mutual distance andr the minimal mutual distance. (1) Mode of the ‘limited’ type: there exist two lengthsm andM such that for any timet: 0〈m≤r≤R≤M〈∞. (2) Modes of ‘oscillating’ type. (2a) Oscillations with respect to the infinite velocities: 0〈r≤R≤M〈∞. There is an infinite number of passages ofr in any neighboorhood of zero. (2b) Oscillations with respect to the infinite distances: 0≤m≤r≤R;r≤M〈∞. This mode corresponds to the ‘almost stable trajectories’ (Khilmy, 1961), it fills certainly a set of measure zero in the phase space (this is not the case for the mode 2a). (3) Modes of the ‘open’ type with a hyperbolic escape. For all of these modesR=0(t) whent→±∞. (3a) Hyperbolic-hyperbolic mode:r=0(t) whent→±∞. (3b) Hyperbolic-elliptic mode: for anyt; 0〈m≤r≤M〈∞, the distancer being between the same two bodies whent→+∞ and whent→−∞. (3c) Capture mode:r=0(t) whent→−∞; 0〈m≤r≤M〈∞ fort≥t 0. (3d) Scattering mode: 0〈m≤r≤M〈∞ fort≤t 0;r=0(t) whent→+∞. (3e) Exchange mode: 0〈m≤r≤M〈∞ for anyt but the distancer is between two different couples of bodies whent→+∞ and whent→−∞. (4) Limit modes and ‘abnormal’ modes. These last modes are rare, they appear for the parabolic escapes, the cases of exact collisions and for asymptotic trajectories.