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Mathematik Für Physiker

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Mathematik Für Physiker Mathematik f¨urPhysiker III Michael Dreher Fachbereich f¨urMathematik und Statistik Universit¨atKonstanz Studienjahr 2012/13 2 Acknowledgements: These are the lecture notes to a third semester course on Mathematics for Physicists, and the author is indebted to Nicola Wurz, Maria Lindauer, Florian Franz, Philip Lindner, Simon Sch¨uz,Samuel Greiner, Christian Schoder, Pascal Gumbsheimer for remarks which helped to improve the presentation, and to the audience for appropriating this huge amount of knowledge. Some Legalese: This work is licensed under the Creative Commons Attribution { Noncommercial { No Derivative Works 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA. I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea{shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. Sir Isaac Newton (1642 { 1727) 1 1As quoted in [3]. 4 Contents I Ordinary Differential Equations7 1 Existence and Uniqueness Results9 1.1 An Introductory Example....................................9 1.2 General Considerations...................................... 12 1.3 The Theorem of Picard and Lindelof¨ ............................ 19 1.4 Comparison Principles...................................... 23 2 Special Solution Methods 25 2.1 Equations with Separable Variables............................... 25 2.2 Substitution and Homogeneous Differential Equations.................... 27 2.3 Power Series Expansions (Or How to Determine the Sound of a Drum)........... 28 2.4 Exact Differential Equations................................... 34 3 Linear Differential Equations and Systems 37 3.1 Linear Differential Equations.................................. 37 3.2 Exp of a Matrix, and (det A)0 .................................. 39 3.3 Linear Systems with General Coefficients........................... 41 3.4 Linear Systems and Equations with Constant Coefficients.................. 45 4 Flows 51 4.1 General Remarks......................................... 51 4.2 Dynamical Systems and Stability................................ 55 4.3 Outlook: The Over Head Pendulum.............................. 61 4.4 Geometric Investigations of Dynamical Systems........................ 65 5 Numerical Methods 73 5.1 Explicit Methods......................................... 73 5.2 Implicit Methods......................................... 77 5.3 Symplectic Methods....................................... 79 6 Boundary Value Problems and Eigenvalues 85 6.1 Introduction............................................ 85 6.2 Solutions to First Order BVPs................................. 87 6.3 Second Order Scalar BVPs................................... 90 6.4 Playing in Hilbert Spaces.................................... 94 6.5 Orthogonal Polynomials..................................... 98 5 6 CONTENTS 6.6 Applications of Orthogonal Polynomials............................ 103 II Complex Analysis (Funktionentheorie) 111 7 Holomorphic Functions 113 7.1 Back to the Roots........................................ 113 7.2 Differentiation.......................................... 116 7.3 Conclusions and Applications.................................. 119 8 Integration 127 8.1 Definition and Simple Properties................................ 127 8.2 The Cauchy Integral Theorem.................................. 130 9 Zeroes, Singularities, Residues 139 9.1 Zeroes of Holomorphic Functions................................ 139 9.2 Singularities............................................ 143 9.3 The Residue Theorem...................................... 147 10 Applications of Complex Analysis 153 10.1 Behaviour of Functions...................................... 153 10.2 The Laplace Transform..................................... 156 10.3 Outlook: Maxwell's Equations in the Vacuum......................... 166 A The Fourier Transform 171 A.1 Some Function Spaces and Distribution Spaces........................ 171 A.2 The Fourier Transform on L1(Rn), S(Rn) and L2(Rn)..................... 173 A.3 The Fourier Transform on S0(Rn)................................ 175 B Core Components 179 Part I Ordinary Differential Equations 7 Chapter 1 Existence and Uniqueness Results 1.1 An Introductory Example We consider a thermodynamical system1 | think of a closed balloon with a certain type of gas in it | and (some of) the thermodynamical quantities are the pressure p, temperature T , specific volume τ, which is the volume per unit mass, density % with %τ = 1 per definition, specific entropy S, which is the entropy per unit mass, specific interior energy e, specific enthalpy i, defined as i = e + pτ. We assume that these quantities do not depend on the space variable x in the balloon, and they do not depend on the time variable t. It turns out that of these 7 quantities (for any fixed system), only two are independent, for instance S and τ. All the other five quantities can be expressed as functions of (S; τ), and these functions depend on the medium under consideration. We now concentrate on the function e = e(S; τ), and one of the key relations of thermodynamics is the formula de = T dS − p dτ; or expressed in more mathematical style, @e @e (S; τ) = T; (S; τ) = −p: (1.1) @S @τ A caloric equation of state2 then is p = f(%; S); p = g(S; τ); where f and g depend on the properties of the medium, and f, g are related to each other because of % = τ −1. A physically reasonable assumption is @f > 0; @% 1This exposition follows [5]. 2kalorische Zustandsgleichung 9 10 CHAPTER 1. EXISTENCE AND UNIQUENESS RESULTS which immediately implies @g < 0: @τ Then we may define c := p@p=@% as sound speed of the medium. Be careful: this formula holds only if p is written as a function of % and S. Another reasonable assumption from physics is g to be convex in τ, hence @2g (S; τ) > 0; @τ 2 and also @g (S; τ) > 0: @S Definition 1.1. A medium is called ideal gas if pτ = RT; with R being a constant depending only on the medium3. Proposition 1.2. Under additional assumptions to appear during the course of the proof, the interior energy of an ideal gas depends only on the temperature T . Quasi{Proof. We start with e = e(S; τ) and (1.1). Then we find @e @e R + τ = RT + τ · (−p) = 0; @S @τ or more in detail @e @e R (S; τ) + τ (S; τ) = 0: (1.2) @S @τ This is a differential equation, and in particular, it is a partial differential equation (PDE), because partial derivatives appear, first order differential equation, because higher order derivatives are absent, @ @ linear differential equation, because the operator R @S + τ @τ is a linear operator. PDEs are hard to investigate, which is the reason why this course concentrates on easier equations, so- called ordinary differential equations (ODE), and now we rely on a flash of inspiration which recommends the ansatz e(S; τ) = h(τ · H(S)); with unknown functions h and H. However, following this way we will never know if all solutions e = e(S; τ) to (1.2) can be expressed by this ansatz. Anyway, plugging the ansatz into (1.2) yields Rh0 · τH0(S) + τh0 · H(S) = 0; and the physical assumptions τ > 0; h0 6= 0 make division possible: RH0(S) + H(S) = 0: (1.3) This is an ordinary differential equation, because no partial derivatives exist. Moreover, it is 3 more precisely: R is the universal gas constant divided by the effective molecular weight of the gas under consideration 1.1. AN INTRODUCTORY EXAMPLE 11 of first order, linear, homogeneous, with constant coefficients. 4 One more physical assumption is H 6= 0. Then we can divide once again, and H0(S) R + 1 = 0; H(S) d 1 ln jH(S)j = − ; dS R Z S0 d Z S0 1 ln jH(S)j dS = − dS; S=1 dS S=1 R 1 ln jH(S )j − ln jH(1)j = − (S − 1); 0 R 0 H(S0) 1 ln = − (S0 − 1); H(1) R 1 S jH(S )j = jH(1)j exp exp − 0 : 0 R R If we take the freedom to introduce a constant C0 2 R, then we can write H(S0) = C0 exp(−S0=R): Incorporating this constant C0 into an updated version of the function h, we then find e(S; τ) = h(τ exp(−S=R)); and the consequences are then @e p = − = −h0(τ exp(−S=R)) exp(−S=R) = −h0(%−1 exp(−S=R)) exp(−S=R): @τ Because of p > 0 everywhere, this gives the necessary condition h0 < 0 on the function h. We also wanted @p(%;S) 00 to have @% > 0, from which we deduce that h > 0. Additionally, @e 1 T = (S; τ) = − h0(τ exp(−S=R)) · τ exp(−S=R): @S R We put y = τ exp(−S=R), and it follows that 1 T = − h0(y)y; R hence T depends on y only. An information from physics is that this dependence is typically monotonically decreasing, and therefore an inverse function exists of the form y = y(T ). We can not express this function as a formula, but we know its existence. Then it follows that e = h(τ exp(−S=R)) = h(y) = h(y(T )); and indeed e depends on T only, but no other second thermodynamic quantity. For completeness, we list the assumptions made: • e has the form e = h(τH(S)), • h0 6= 0 and H 6= 0 (which means that these functions take nowhere the value zero), 4 In comparison: (1.2) is also linear and homogeneous, but has variable coefficients. 12 CHAPTER 1. EXISTENCE AND UNIQUENESS RESULTS • the function T = T (y) is strictly monotone. If the last condition seems to restrictive, we could it replace it by the condition that T has only a finite number of intervals of monotonicity, and consider only such systems where T stays in the same interval of monotonicity. Corollary 1.3. Also the sound speed depends only on the temperature, because of @p @ c2 = (%; S) = − h0(%−1) exp(−S=R) exp(−S=R) @% @% = h00(%−1 exp(−S=R))(exp(−S=R))2%−2 = h00(y)y2 = h00(y(T )) · (y(T ))2: Let us take a step back, go to the meta-level, and have a look what we have done so far.
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