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The Lazy UniverseAn Introduction to the Principle of Least Action$
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Jennifer Coopersmith

Print publication date: 2017

Print ISBN-13: 9780198743040

Published to Oxford Scholarship Online: June 2017

DOI: 10.1093/oso/9780198743040.001.0001

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D’Alembert’s Principle

D’Alembert’s Principle

Chapter:
(p.88) 5 D’Alembert’s Principle
Source:
The Lazy Universe
Author(s):

Jennifer Coopersmith

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198743040.003.0005

It is explained how the mysterious Principle of Virtual Work in statics is extended to the even more mysterious Principle of d’Alembert’s in dynamics. This is achieved by d’Alembert’s far-sighted stratagem: considering a reversed massy acceleration as an inertial force. A worked example is given (the half-Atwood machine or “black box”). Some counter-intuitive aspects are made intuitive by more examples: the Pluto-Charon system of orbiting planets; Newton’s and then Mach’s explanation of Newton’s bucket. Also, it is demonstrated that the law of the conservation of energy actually follows from d’Alembert’s Principle. The reader is alerted to the astoundingly fundamental nature of d’Alembert’s Principle. It is the cornerstone of classical, relativistic, and quantum mechanics. As Lanczos writes: “All the different principles of mechanics are merely mathematically different formulations of d’Alembert’s Principle”.

Keywords:   d’Alembert’s Principle, dynamics, inertial force, Newton’s bucket, Mach, Pluto and Charon, black box, conservation of energy

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