On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms

Abstract

We reconsider the problem of the Hamiltonian interpolation of symplectic mappings. Following Moser's scheme, we prove that for any mapping ψ_ε, analytic and ε-close to the identity, there exists an analytic autonomous Hamiltonian system, H_ε such that its time-one mapping Φ_Hε differs from ψ_ε 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 orders to integrate a Hamiltonian systemK, one actually follows “exactly,” namely within the computer roundoff error, the trajectories of the interpolating Hamiltonian H_ε, or equivalently of the rescaled Hamiltonian K_ε=ε^-1H_ε, which differs fromK, but turns out to be ε⁵ close to it. Special attention is devoted to numerical integration for scattering problems.