lecture notes by Michael Stone - Mathematics for Physics I...

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Unformatted text preview: Mathematics for Physics I A set of lecture notes by Michael Stone PIMANDER-CASAUBON Alexandria Florence London ii Copyright c 2001,2002 M. Stone. All rights reserved. No part of this material can be reproduced, stored or transmitted without the written permission of the author. For information contact: Michael Stone, Loomis Laboratory of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801, USA. Preface These notes were prepared for the first semester of a year-long mathematical methods course for begining graduate students in physics. The emphasis is on linear operators and stresses the analogy between such operators acting on function spaces and matrices acting on finite dimensional spaces. The op-erator language then provides a unified framework for investigating ordinary and partial differential equations, and integral equations. The mathematical prerequisites for the course are a sound grasp of un-dergraduate calculus (including the vector calculus needed for electricity and magnetism courses), linear algebra (the more the better), and competence at complex arithmetic. Fourier sums and integrals, as well as basic ordinary differential equation theory receive a quick review, but it would help if the reader had some prior experience to build on. Contour integration is not required. iii iv PREFACE Contents Preface iii 1 Calculus of Variations 1 1.1 What is it good for? . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 The functional derivative . . . . . . . . . . . . . . . . . 2 1.2.2 The Euler-Lagrange equation . . . . . . . . . . . . . . 3 1.2.3 Some applications . . . . . . . . . . . . . . . . . . . . . 4 1.2.4 First integral . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . 10 1.3.1 One degree of freedom . . . . . . . . . . . . . . . . . . 11 1.3.2 Noethers theorem . . . . . . . . . . . . . . . . . . . . 15 1.3.3 Many degrees of freedom . . . . . . . . . . . . . . . . . 18 1.3.4 Continuous systems . . . . . . . . . . . . . . . . . . . . 19 1.4 Variable End Points . . . . . . . . . . . . . . . . . . . . . . . . 29 1.5 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . 34 1.6 Maximum or Minimum? . . . . . . . . . . . . . . . . . . . . . 38 1.7 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . . 40 2 Function Spaces 49 2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.1.1 Functions as vectors . . . . . . . . . . . . . . . . . . . 50 2.2 Norms and Inner Products . . . . . . . . . . . . . . . . . . . . 51 2.2.1 Norms and convergence . . . . . . . . . . . . . . . . . . 51 2.2.2 Norms from integrals . . . . . . . . . . . . . . . . . . . 53 2.2.3 Hilbert space . . . . . . . . . . . . . . . . . . . . . . . 55 2.2.4 Orthogonal polynomials . . . . . . . . . . . . . . . . . 63 2.3 Linear Operators and Distributions . . . . . . . . . . . . . . . 68 v vi...
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This note was uploaded on 01/29/2008 for the course PHYS 598 taught by Professor Stone during the Fall '07 term at University of Illinois at Urbana–Champaign.

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lecture notes by Michael Stone - Mathematics for Physics I...

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