arXiv subjects

Gaetano Zampieri

Publications and source records attributed to Gaetano Zampieri.

20 records · Page 2Linked to original sources

arXiv2012

Complete Integrability for Hamiltonian Systems with a Cone Potential

It is known that, if a point in $R^n$ is driven by a bounded below potential $V$, whose gradient is always in a closed convex cone which contains no lines, then the velocity has a finite limit as time goes to $+\infty$. The components of the asymptotic velocity, as functions of the initial data, are trivially constants of motion. We find sufficient conditions for these functions to be $C^k$ ($2\le k \le+\infty$) first integrals, independent and pairwise in involution. In this way we construct a large class of completely integrable systems. We can deal with very different asymptotic behaviours of the potential and we have persistence of the integrability under any small perturbation of the potential in an arbitrary compact set.

nlin.SI↗

arXiv2012

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

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

nlin.SI↗