Classical Motion in Angular Potentials

Abstract.

Let V be a function independent of |x| of class C ² on \({\bf R}^n \setminus \{0\}\), \(n \geq 2\), and define \(Cr = \{\omega\in S^{n-1} : \nabla V(\omega)=0\}\). We prove that if Cr is a totally disconnected subset of S ⁿ⁻¹ and if x(·) is a solution of Newton’s equation \(\ddot{x}(t) = -\nabla V\) which is unbounded for positive times, then \({\rm lim}_{t\rightarrow\infty}x(t)/|x(t)|\) exists and belongs to Cr.