On the Hamiltonian Interpolation of Near to the Identity Symplectic Mappings

We reconsider the problem of the Hamiltonian interpolation of symplectic mappings. Following Moser's scheme, we prove that for any mapping PSI(epsilon), analytic and epsilon-close to the identity, there exists an analytic autonomous Hamiltonian system, H(epsilon) such that its time-one mapping PHI(Hepsilon) differs from PSI(epsilon) by a quantity exponentially small in 1/epsilon. This result is applied, in particular, to the problem of numerical integration of Hamiltonian systems by symplectic algorithms; it turns out that, when using an analytic symplectic algorithm of order s to integrate a Hamiltonian system K, one actually follows ''exactly,'' namely within the computer roundoff error, the trajectories of the interpolating Hamiltonian H(epsilon), or equivalently of the rescaled Hamiltonian K(epsilon) = epsilon-1 H(epsilon), which differs from K, but turns out to be epsilon(s) close to it. Special attention is devoted to numerical integration for scattering problems.