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Analytical Mechanics for Relativity and Quantum Mechanics$
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Oliver Johns

Print publication date: 2005

Print ISBN-13: 9780198567264

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198567264.001.0001

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Generating Functions

Generating Functions

(p.434) 18 Generating Functions
Analytical Mechanics for Relativity and Quantum Mechanics

Oliver Johns

Oxford University Press

In the previous chapter, several conditions were given that could be used for testing transformations to see whether they were canonical or not. But no methods, other than trial and error, were given for actually creating the canonical transformations to be tested. In this chapter, methods for creating transformations that will automatically be canonical are presented. Canonical transformations can be created by first choosing what are called generating functions. Using one of these generating functions in the formalism to be described will generate a transformation that will be canonical by construction. The generating functions can be quite general, leading to a wide selection of possible canonical transformations. This chapter also discusses proto-generating functions, examples of generating functions, mixed generating functions, simple generating functions, traditional generating functions, standard form of extended Hamiltonian recovered, differential canonical transformations, active canonical transformations, phase-space analog of Noether theorem, and Liouville theorem.

Keywords:   canonical transformations, generating functions, Noether theorem, Liouville theorem, mixed generating functions, proto-generating functions, simple generating functions

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