Asymptotics of trajectories for cone potentials (Q1072183)

A trajectory of particle under potential field in \({\mathbb{R}}^ n\) is called asymptotically uniform if it asymptotically coincides with a free particle one: \(q(t)=p_{\pm}t+q_{\pm}+O(1)\) for \(t\to \pm \infty\). A potential V is called a cone potential if a closed cone spanned by \(\nabla_ qV\), \(q\in {\mathbb{R}}^ n\), does not contain a straight line. For a cone potential asymptotic velocities \(p_{\pm}\) always exist (see the author's earlier work). A necessary and sufficient condition for almost all trajectories to be asymptotically uniform is found here for cone potential under some general assumptions.