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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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Geometric theory of ordinary differential equations

Geometric theory of ordinary differential equations

Chapter:
(p.33) 6 Geometric theory of ordinary differential equations
Source:
Advanced Mechanics
Author(s):

S. G. Rajeev

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199670857.003.0006

All physics is based on differential equations. Ordinary differential equations appear in mechanics. They can be thought of as the integral curves of a vector field on a manifold, the Phase Space. Vector fields are derivations of the algebra of functions. Of special interest is a vector field near a fixed point. Stable and unstable directions are defined. In an exercise, the dynamics near a double zero on the real line is developed. Another exercise shows Lorenz's butterfly: a glimpse of chaos to come.

Keywords:   vector field, integral curves, fixed point, Lorenz's butterfly

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