Integrable Hamiltonians with exponential potential (Q1069526)

The author considers completely integrable Hamiltonian systems. Given a Hamiltonian H(p,q) defined on phase space \({\mathbb{R}}^{2n}\), the equations of motion are \(\dot q_ i=\partial H/\partial p_ i\), \(\dot p_ i=- \partial H/\partial q_ i\). Such a system is called asymptotically free if the limits \(F_ i(p,q)=\lim_{t\to \infty}p_ i(t)\) or \(G_ i(p,q)=\lim_{t\to -\infty}p_ i(t)\) exist for any trajectory \(\{\) (p(t),q(t)): \((p,q)=(p(0),q(0))\}\). The author shows that such a system is completely integrable. Special reference is made to the Toda lattice, and the author establishes that this system is a member of a large class of Hamiltonians with exponential potential that generate an asymptotically free dynamics.