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Mechanistic Images in Geometric FormHeinrich Hertz's 'Principles of Mechanics'$
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Jesper Lützen

Print publication date: 2005

Print ISBN-13: 9780198567370

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198567370.001.0001

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Cyclic and conservative systems

Cyclic and conservative systems

Chapter:
(p.225) 20 Cyclic and conservative systems
Source:
Mechanistic Images in Geometric Form
Author(s):

JESPER LÜTZEN

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198567370.003.0020

The most interesting example of a system acted on by forces is the so-called conservative system in which a force function exists. This is a system consisting of a visible system coupled to a special kind of hidden system. In order to deal with such systems, Heinrich Hertz first introduced the concept of an adiabatic cyclic system, which was intended to play the role as the second (hidden) subsystem in a conservative system. In accordance with his overall approach, Hertz defined cyclic coordinates geometrically. Since the energy is defined by the same quadratic form as the line element, this definition is equivalent to the usual definition of a cyclic coordinate as a coordinate that does not enter into the expression of the energy of the system. This chapter discusses adiabatic cyclic systems, conservative systems, cases where hidden non-holonomic connections are not allowed, and the approximative character of cyclic and conservative systems.

Keywords:   cyclic systems, cyclic coordinates, adiabatic cyclic systems, forces, energy, conservative systems, hidden non-holonomic connections

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