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Advanced MechanicsFrom Euler's Determinism to Arnold's Chaos$
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S. G. Rajeev

Print publication date: 2013

Print ISBN-13: 9780199670857

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199670857.001.0001

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Discrete time

Discrete time

Chapter:
(p.117) 15 Discrete time
Source:
Advanced Mechanics
Author(s):

S. G. Rajeev

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199670857.003.0015

Whether time is discrete at the fundamental level is still unknown. But for computational purposes, it is very useful to have a form of mechanics in which time increases in finite steps. Each step must be a canonical transformation. Using the Poincare-Birkhoff-Witt lemma (also called the Baker-Campbell-Hausdorff formula) this chapter derives first and second order symplectic integrators that allow a discrete time formulation of mechanics. The example of the standard map of Chiriikov (discrete time form of the pendulum) gives the first example of chaos. Arnold's cat map is introduced in an exercise.

Keywords:   symplectic integrator, Poincare-Birkhoff-Witt lemma, Baker-Campbell-Hausdorff formula, Chiriikov, chaos

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