Variational integrators for orbital problems using frequency estimation

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Abstract

In this work, we present a new derivation of higher order variational integration methods that exploit the phase lag properties for numerical integrations of systems with oscillatory solutions. More specifically, for the derivation of these integrators, the action integral along any curve segment is defined using a discrete Lagrangian that depends on the endpoints of the segment and on a number of intermediate points of interpolation. High order integrators are then obtained by writing down the discrete Lagrangian at any time interval as a weighted sum of the Lagrangians corresponding to a set of the chosen intermediate points. The respective positions and velocities are interpolated using trigonometric functions. The methods derived this way depend on a frequency, which in general needs to be accurately estimated. The new methods, which improve the phase lag characteristics by re-estimating the frequency at every time step, are presented and tested on the general N-body problem as numerical examples.

Keywords

Variational integrators Discrete variational mechanics Highly oscillatory problems Phase lag analysis General N-body problem 

Mathematics Subject Classification (2010)

49Mxx 65Kxx 

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Modelling and Simulation Centre, MACEUniversity of ManchesterManchesterUK
  2. 2.Chair of Applied DynamicsUniversity of Erlangen-NurembergErlangenGermany

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