Abstract
We reconsider the problem of the Hamiltonian interpolation of symplectic mappings. Following Moser's scheme, we prove that for any mapping ψe, analytic and e-close to the identity, there exists an analytic autonomous Hamiltonian system, He such that its time-one mapping ΦHe differs from ψe by a quantity exponentially small in 1/e. 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 He, or equivalently of the rescaled Hamiltonian Ke=e-1He, which differs fromK, but turns out to be e5 close to it. Special attention is devoted to numerical integration for scattering problems.
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