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Analytical Mechanics for Relativity and Quantum Mechanics$
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Oliver Johns

Print publication date: 2011

Print ISBN-13: 9780191001628

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780191001628.001.0001

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Introduction to Hamiltonian Mechanics

Introduction to Hamiltonian Mechanics

Chapter:
(p.70) 4 Introduction to Hamiltonian Mechanics
Source:
Analytical Mechanics for Relativity and Quantum Mechanics
Author(s):

Oliver Davis Johns

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780191001628.003.0004

This chapter argues that Hamiltonian mechanics is a much better base from which to build more advanced methods. The Hamilton equations have an elegant symmetry that the Lagrange equations lack. The Hamiltonian function is also used to write the Schroedinger equation of quantum mechanics. The differences between the Lagrange and Hamilton equations result mainly from the different variable sets in which they act. The Lagrangian variable set is the set of generalised coordinates and velocities, whereas the Hamiltonian set is the set of generalised coordinates and momenta. The transformation from Lagrange to Hamilton equations is a Legendre transformation, in which the Lagrangian function of the Lagrangian variable set is to be replaced by the Hamiltonian function of the phase-space variable set.

Keywords:   quantum mechanics, Hamiltonian mechanics, Hamilton equations, Lagrange equations, Schroedinger equation

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