Parabolic orbits in Celestial Mechanics: a functional‐analytic approach

Abstract

We prove the existence of half-entire parabolic solutions, asymptotic to a prescribed central configuration, for the equation \begin{equation*} \ddot{x} = \nabla U(x) + \nabla W(t,x), \qquad x \in \mathbb{R}^{d}, \end{equation*} where $d \geq 2$, $U$ is a positive and positively homogeneous potential with homogeneity degree $-\alpha$ with $\alpha\in\mathopen{]}0,2\mathclose{[}$, and $W$ is a (possibly time-dependent) lower order term, for $\vert x \vert \to +\infty$, with respect to $U$. The proof relies on a perturbative argument, after an appropriate formulation of the problem in a suitable functional space. Applications to several problems of Celestial Mechanics (including the $N$-centre problem, the $N$-body problem and the restricted $(N+H)$-body problem) are given.