(Bi)-hyperbolic and (bi)-parabolic motions in the restricted (N+1)-body problem
par
Danjon
Paris
According to Chazy, the final motion of the restricted (N+1)-body problem has four possibilities: bounded, hyperbolic, parabolic and oscillatory. When the motion is hyperbolic or parabolic, the massless body will go to infinity along a definite asymptotic direction with a finite limiting energy. Then there are two basic questions: First given any initial time and position, as well as the asymptotic direction and limiting energy when time goes to positive infinity, is there a corresponding hyperbolic or parabolic motion realizing it; Second given any asymptotic directions and limiting energy, when time goes to both negative and positive infinity, is there a corresponding bi-hyperbolic or bi-parabolic motion realizing it? In this talk, we will report some of our progress in these two questions.
Alain Albouy, Alain Chenciner, Jacques Laskar