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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

(p.33) 6 Geometric theory of ordinary differential equations
Advanced Mechanics

S. G. Rajeev

Oxford University Press

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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