Complete integrability for Hamiltonian systems with a cone potential

Abstract

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 +infinity. 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<k<infinity) 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

Liouville-Arnold Integrability for Scattering Under Cone Potentials

“…If V decays exponentially at infinity (see a later section for precise statements), we give Gronwall-like estimates on the growth of the derivatives of q with respect to the initial data. If instead V is assumed convex, then a simple monotonicity condition on the Hessian matrix of V enables us to build a Liapunov function that checks the growth of the derivatives of q without special decay requirements on V. In both settings it is proved, again in [GZ1], that the components of p ∞ , in addition to being differentiable, are independent and in involution on M, and the range of p ∞ (on M) is exactly the interior of the dual cone C * . In particular, the Hamiltonian system is integrable if it is restricted to the open, nonempty, invariant set M. II.…”

“…The conjecture, suggested by Gutkin in [Gu1], is false without further restrictions on the potential, though. In particular, the asymptotic velocity may not even be a continuous function of the initial data (see [GZ1], Section 3), and sometimes there are geometrical obstructions to the global independence of its components, even if they happened to be smooth (see [GZ3], Sections 1 and 5).…”

“…In the papers [GZ1,2,3] the authors develop a theory of the C k (2 ≤ k ≤ +∞) regularity of the asymptotic velocity and Liouville-Arnold integrability for cone potentials, whose main points we are going to sketch here.…”

“…Under very general and natural hypotheses on the potential (made explicit in [GZ3]), M is also open and nonempty. The bulk of [GZ1] is devoted to proving that the asymptotic velocity p ∞ is a smooth (C k , 2 ≤ k ≤ +∞) function of the initial data on M, if suitable additional hypotheses are made on V. The starting point is the integral expression for the asymptotic velocity:…”

“…In particular, the Hamiltonian system is integrable if it is restricted to the open, nonempty, invariant set M. II. In [GZ1,3] we provide examples where M does not coincide with the whole phase space R n × R n . This can happen either for a trivial reason such as the presence of an equilibrium (∇V(q) = 0), or for subtler reasons.…”