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 "There is a simple and powerful integrator (the 'asynchronous leap-frog \
integrator') for the general ordinary differential equation which is strictly \
reversible in the sense that each discrete trajectory can be reconstructed  \
exactly if only the final state is known. It is an interesting question how \
such a behavior manifests itself in situations where the physical process \
represented by the differential equation is not reversible. We are going to \
study this now by considering a simple example.\n \nA well-known irreversible \
physical process is the emptying of a water-filled  bucket due to a hole in \
the bottom. The water level ",
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