On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms |
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Authors: | Giancarlo Benettin Antonio Giorgilli |
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Affiliation: | (1) Dipartimento di Matematica Pura e Applicata, GNFM (CNR) and INFM, Università di Padova, 35131 Padova, Italy;(2) Dipartimento di Matematica and GNFM (CNR), Università di Milano, 20133 Milano, Italy |
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Abstract: | We reconsider the problem of the Hamiltonian interpolation of symplectic mappings. Following Moser's scheme, we prove that for any mapping , analytic and -close to the identity, there exists an analytic autonomous Hamiltonian system, H such that its time-one mapping H differs from by a quantity exponentially small in 1/. 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, or equivalently of the rescaled Hamiltonian K=-1H, which differs fromK, but turns out to be 5 close to it. Special attention is devoted to numerical integration for scattering problems. |
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Keywords: | Hamiltonian systems symplectic mappings symplectic integration algorithms perturbation theory |
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