A new form of equations of motion in celestial mechanics (Q1363894)

The aim is to describe some systems of celestial mechanics (or more general dynamical systems that can be expressed in terms of a special second order system of differential equations \({\mathbf x}''= {\mathbf f} ({\mathbf x}))\) by using a set of variables that can be separated into two groups with bounded and unbounded variation. Assuming that the system possesses a global solution which is used as a reference orbit, the author uses a geometrical reason to introduce new variables and to derive differential equations in these variables. Some stability results for a periodic reference orbit are presented. It must be noticed that the main results of the paper are only stated, sometimes in an informal way, and no precise proofs are given. So, in theorem 5 which deals with the gravitational motion of a system of \(n\)-bodies around their center of mass, taking as reference orbit of the whole system a set of fixed elliptic orbits for each body, the author states the following global stability result: ``any movements starting in a sufficiently small vicinity of the elliptic orbits remain within the small vicinities of these orbits''.