We consider classical potential scattering. If at energy E no orbit is trapped, the Hamiltonian dynamics define an integer-valued topological degree deg(E) ≤ 1. This is calculated explicitly for all potentials, and exactly integers ≤1 are shown to occur for suitable potentials.
The non-trapping condition is restrictive in the sense that for a bounded potential it is shown to imply that boundary of Hill's region in configuration space is either empty or homeomorphic to a sphere.
However, in many situations one can decompose a potential into a sum of non-trapping potentials with a non-trivial degree and embed symbolic dynamics of multi-obstacle scattering. This comprises a large number of earlier results, obtained by different authors on multi-obstacle scattering.