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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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Canonical Transformations

Canonical Transformations

(p.429) 19 Canonical Transformations
Analytical Mechanics for Relativity and Quantum Mechanics

Oliver Davis Johns

Oxford University Press

This chapter describes canonical transformations as the most general phase-space transformations that preserve the extended Hamilton equations. There are several equivalent definitions of canonical transformations; three of them are presented here: the Poisson bracket condition, the direct condition, and the Lagrange bracket condition. Each of these three conditions has two forms, a long one that is written out in terms of partial derivatives and a symplectic one consisting of a single matrix equation. The chapter begins with the long form of the Lagrange bracket condition, and after introducing some necessary notation, derives the long and symplectic forms of all three. The definition of canonical transformation includes the Lorentz transformation of special relativity, since we are now operating in an extended phase space.

Keywords:   canonical transformations, phase-space transformations, extended Hamilton equations, Poisson bracket condition, direct condition, Lagrange bracket condition

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