TL;DR
This paper investigates the properties of Liouville operators in symplectic integrators applied to a one-dimensional harmonic oscillator, revealing limitations related to time step size and behavior of coordinate and momentum.
Contribution
It derives effective Liouville operators for first- and second-order symplectic integrators specific to the harmonic oscillator, highlighting their constraints.
Findings
Liouville operators are valid only for time steps less than two
Coordinate and momentum magnitudes increase monotonically at large time steps
Effective operators are explicitly constructed for the harmonic oscillator
Abstract
Effective Liouville operators of the first- and the second-order symplectic integrators are obtained for the one-dimensional harmonic-oscillator system. The operators are defined only when the time step is less than two. Absolute values of the coordinate and the momentum monotonically increase for large time steps.
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