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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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The Calculus of Variations

The Calculus of Variations

(p.92) 5 The Calculus of Variations
Analytical Mechanics for Relativity and Quantum Mechanics

Oliver Davis Johns

Oxford University Press

This chapter introduces the calculus of variations in the context of finite-dimensional configuration spaces. Mastery of this relatively simple form of the theory will provide the background required to study more advanced topics such as the variations of fields in the complex spaces of quantum field theory. The presentation of the calculus of variations here uses the so-called General Parametric Method. In it, a path is specified parametrically, by letting each of its coordinates be a function of some monotonically varying, but initially unspecified, parameter. This method contrasts with some other textbooks in which one of the coordinates is used as the parameter, and the other variables are made functions of it. The two methods are compared in detail in the sections that follow.

Keywords:   calculus of variations, configuration spaces, quantum field theory, General Parametric Method, complex spaces

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