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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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Mathematics and physics preliminaries: of hills and plains and other things

Mathematics and physics preliminaries: of hills and plains and other things

Chapter:
(p.31) 3 Mathematics and physics preliminaries: of hills and plains and other things
Source:
The Lazy Universe
Author(s):

Jennifer Coopersmith

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198743040.003.0003

The link between mathematics and physics is explained, and how the concepts “coordinates,” “generalized coordinates,” “time,” and “space” have evolved, starting with Galileo. It is also shown that “degrees of freedom” is a slippery but crucial idea. The important developments in “space research”, from Pythagoras to Riemann, are sketched. This is followed by the motivations for finding a flat region of “space”, and for Riemann’s invariant interval. A careful explanation of the three ways of taking an infinitesimal step (actual, virtual, and imperfect) is given. The programme of the Calculus of Variations is described and how this requires a virtual variation of a whole path, a path taken between fixed end-states. This then culminates in the Euler-Lagrange Equations or the Lagrange Equations of Motion. Along the way, the ideas virtual displacement and extremum are explained.

Keywords:   generalized coordinates, time, degrees of freedom, Riemann, invariant interval, Calculus of Variations, Euler-Lagrange Equations, virtual displacement, extremum, whole path

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