Reducing scattering problems under cone potentials to normal form by global canonical transformations

Abstract

We introduce a class of Hamiltonian scattering systems which can be reduced to the "normal form"Ṗ = 0,Q = P , by means of a global canonical transformation (P, Q) = A(p, q), p, q ∈ I R n , defined through asymptotic properties of the trajectories.These systems are obtained requiring certain geometrical conditions oṅ p = −∇V(q),q = p, where V is a bounded below "cone potential", i.e., the force −∇V(q) always belongs to a closed convex cone which contains no straight lines.We can deal with very different asymptotic behaviours of the potential and the potential can undergo small perturbations in any arbitrary compact set without losing the existence and the properties of A.

Liouville-Arnold integrability for scattering under cone potentials

“…All the previous examples share the "finite sum" form, that gives rise to polyhedral cones C and C * . This is not an essential feature of the theory, as we show in [GZ2], where we prove the C ∞ -integrability of the system in three dimensions whose potential is…”

Liouville-Arnold Integrability for Scattering Under Cone Potentials

“…This is not an essential feature of the theory, as we show in [GZ2], where we prove the C ∞ -integrability of the system in three dimensions whose potential is…”

“…The paper [GZ2] shows that if we restrict to potentials with fast enough decay at infinity, then the motion of the particle q is asymptotically rectilinear uniform: q(t) = a ∞ + p ∞ t + o(1) as t → +∞, and the asymptotic data (p ∞ , a ∞ ), as functions of the initial data, define a global canonical diffeomorphism ("asymptotic map") A: (p, q) → (P, Q), which brings the original Hamiltonian system (1.1) into the normal form:…”

A class of integrable Hamiltonian systems including scattering of particles on the line with repulsive interactions

“…In [GZ2] we somewhat specialize our hypotheses (the ones connected with point 2 only) to permit the reduction of (1.1) to the normal form…”

“…where a ∞ (p, q), the asymptotic phase, is given by a ∞ (p, q) := lim t→+∞ q(t, p, q) − t p ∞ (p, q) , (1.7) (this limit always exists within the assumptions of [GZ2]).…”