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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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The simple pendulum

The simple pendulum

Chapter:
(p.11) 3 The simple pendulum
Source:
Advanced Mechanics
Author(s):

S. G. Rajeev

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199670857.003.0003

The simple pendulum is a simple mechanical system with a surprising depth. It can be solved in terms of Jacobi elliptic functions. A simple physical argument shows that these functions are doubly periodic when continued into complex time. As an aside the theory of elliptic curves is developed: addition formula, the Gauss-Landen transformation. The arithmetic-geometric mean is defined by an iterative property. It allows the period of a pendulum to be calculated accurately without performing integrals.

Keywords:   pendulum, jacobi functions, elliptic curves, arithmetic-geometric mean, Gauss-Landen transformation

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