On the Hamiltonian Interpolation of Near to the Identity Symplectic Mappings with Application to Symplectic Integration Algorithms

We reconsider the problem of the Hamiltonian interpolation of symplectic mappings. Following Moser's scheme, we prove that for any mapping ψε, analytic and ε-close to the identity, there exists an analytic autonomous Hamiltonian system, Hε such that its time-one mapping ΦHε differs from ψε by a quantity exponentially small in 1/ε. 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 Hε, or equivalently of the rescaled Hamiltonian Kε=ε-1Hε, which differs from K, but turns out to be ε5 close to it. Special attention is devoted to numerical integration for scattering problems.