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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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Lagrangian Mechanics

Lagrangian Mechanics

Chapter:
(p.107) 6 Lagrangian Mechanics
Source:
The Lazy Universe
Author(s):

Jennifer Coopersmith

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198743040.003.0006

It is demonstrated how d’Alembert’s Principle can be used as the basis for a more general mechanics – Lagrangian Mechanics. How this leads to Hamilton’s Principle (the Principle of Least Action) is shown mathematically and in words. It is further explained why Lagrangian Mechanics is so general, why forces of constraint may be ignored, and how external conditions lead to “curved space.” Also, it is explained why the Lagrangian, L, has the form L = TV (where T is the kinetic energy and V is the potential energy), and why T is in “quadratic form” (T = 1/2mv2). It is shown how Noether’s Theorem leads to a more fundamental definition of energy and links the conservation of energy to the homogeneity of time. The ingenious Lagrange multipliers are explained, and also generalized forces and generalized coordinates.

Keywords:   L = T − V, T = 1/2mv2, quadratic form, Noether’s Theorem, Lagrange multipliers, generalized forces, generalized coordinates, curved space, Hamilton’s Principle

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