We introduce a class of Hamiltonian scattering systems which can be reduced to the \it normal form'' \rm \ $$\dot P=0\,, \qquad \dot Q=P\,,$$ by means of a global \it canonical transformation \rm $\ (P,Q)=\A(p,q),\ p,q\in \Rfi^n\,$, defined through \it asymptotic \rm properties of the trajectories. These systems are obtained requiring certain geometrical conditions on $$\dot p=-\nabla \V (q)\,,\qquad \dot q=p\,,$$ where $\V$ is a bounded below \it cone potential''\rm, i.e. the force $-\nabla \V (q)$ 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 $\A$.

### 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 \it normal form'' \rm \ $$\dot P=0\,, \qquad \dot Q=P\,,$$ by means of a global \it canonical transformation \rm $\ (P,Q)=\A(p,q),\ p,q\in \Rfi^n\,$, defined through \it asymptotic \rm properties of the trajectories. These systems are obtained requiring certain geometrical conditions on $$\dot p=-\nabla \V (q)\,,\qquad \dot q=p\,,$$ where $\V$ is a bounded below \it cone potential''\rm, i.e. the force $-\nabla \V (q)$ 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 $\A$.
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Hamiltonian scattering systems, cone potential, global canonical transformation defined through asymptotic limits
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11562/393340
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