Spiral motion in classical mechanics

We present various models in classical mechanics which exhibit 'exotic' orbits. We give an example of a smooth $|\mathbf{r}|$-independent potential
$V$ in dimension three, which exhibits an orbit that spirals as time goes to infinity. This kind of orbits cannot occur for this class of potentials in dimension two [4] or, see below, if ${Cr}=\{\omega\in S^{n-1}:\nabla V(\omega)=0\}$, $n\geq 3$, is totally disconnected. In addition, for each $\mu>2$ we give an example of a potential of the form $V(r,\theta)=O(r^{-\mu})$, in two dimensions, which is not radially symmetric and has a zero-energy orbit that escapes towards infinity in spirals. Zero energy orbits escaping towards infinity in spirals cannot occur for radial potentials with the same rate of decay.