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Lectures on Ordinary Differential Equations

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Lectures on Ordinary Differential Equations DRAFT 1 Lectures on Ordinary Differential Equations (Oxford Physics Papers CP3/4) Alexander A. Schekochihiny The Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, UK Merton College, Oxford OX1 4JD, UK (compiled on 5 February 2021) These are the notes for my lectures on Ordinary Differential Equations for 1st-year undergraduate physicists, taught since 2018 as part of Paper CP3 at Oxford. They also include lectures on Normal Modes (part of Paper CP4), taught sunce 2021. I will be grateful for any feedback from students, tutors or (critical) sympathisers. The latest, up-to-date version of this file is available at http://www-thphys.physics.ox.ac.uk/ people/AlexanderSchekochihin/ODE/2018/ODELectureNotes.pdf. As typos, errors, and better ways of explaining things are found, I will update these notes in real time as lectures proceed. Please check back here for updates (the date tag of the version that you are reading appears above). Below I give references to some relevant bits of various books and online lecture notes. Having access to these materials is not essential for being able to follow these lectures, but reading (these or other) books above and beyond your lecturer's notes is essential for you to be able to claim that you have received an education. The problem sets suggested for your tutorials are at the end of this document. They too will be updated in real time if errors are found or new/better questions arise. CONTENTS 1. The Language of the Game 5 1.1. What is an ODE?5 1.2. Integral Curves and the Cauchy Problem8 1.3. Existence and Uniqueness9 1.4. Parametric Solutions 10 2. Some Methods for Solving First-Order ODEs 12 2.1. Equations in Full Differentials 12 2.1.1. Integrating Factor 13 2.2. Separable Equations 14 2.2.1. Separable Equations and Composite Solutions 14 2.2.2. Reduction to Separable Form by Linear Change of Variables 16 2.3. Homogeneous Equations 17 2.3.1. Reduction to Homogeneous Form by Linear Change of Variables 18 2.4. Linear Equations 19 2.4.1. Solution via Integrating Factor 19 2.4.2. Solution via Variation of Constant 20 2.4.3. CF + PI 21 y E-mail: [email protected] 2 A. A. Schekochihin 2.5. Bernoulli Equations 21 2.6. Riccati Equations 22 2.7. Equations Unresolved With Respect to Derivative 22 2.7.1. Cauchy Problem 23 2.7.2. Solution via Introduction of Parameter 23 3. The Language of the Game: Going to Higher Order 25 3.1. Mathematical Pendulum 25 3.2. Laws of Motion 26 3.3. All ODEs Are (Systems of) First-Order ODEs 27 3.4. Existence and Uniqueness 28 3.5. Phase Space and Phase Portrait 29 3.5.1. Linear Pendulum 30 3.5.2. Nonlinear Pendulum 31 3.5.3. Local Linear Analysis 33 3.5.4. Damped Pendulum (Introduction to Dissipative Systems) 33 4. Linear ODEs: General Principles 35 4.1. Existence and Uniqueness Theorem for Linear Equations 35 4.2. Superposition Principle 36 4.2.1. Superposition Principle for Inhomogeneous Equations 36 4.3. General Solution of Homogeneous Equations 37 4.3.1. (In)dependent Somewhere|(In)dependent Everywhere 38 4.3.2. The Fundamental Matrix and the Wronskian 38 4.3.3. How to Construct a Fundamental System 39 4.3.4. How to Construct the Solution 39 4.4. General Solution of Inhomogeneous Equations 41 4.5. Green's Function 42 4.6. Buy One Get One Free 44 4.6.1. Tips for Guessing Games 46 5. Second-Order Linear ODE with Constant Coefficients 47 5.1. Homogeneous Equation 47 5.1.1. Damped Oscillator 48 5.1.2. Homogeneous Equation: Degenerate Case 49 5.1.3. Above and Beyond: n-th-Order Homogeneous Equation 50 5.1.4. Scale-Invariant (Euler's) Equation 50 5.2. Inhomogeneous Equation 51 5.2.1. Some Tips for Finding Particular Solutions 52 5.2.2. Above and Beyond: Quasipolynominals and n-th-Order Inhomoge- neous Equation 53 5.3. Forced Oscillator 54 5.3.1. Resonance 55 5.3.2. Energy Budget of Forced Oscillator 57 5.4. (Nonlinear) Digression: Rapidly Oscillating Force 58 6. Systems of Linear ODEs with Constant Coefficients 60 6.1. Diagonalisable Systems With No Degenerate Eigenvalues 61 6.1.1. General Solution of Inhomogeneous Equation 64 6.2. Hermitian Systems 66 6.3. Non-Hermitian Systems 67 6.3.1. Solution by Triangulation 68 6.3.2. Proof of Schur's Triangulation Theorem 69 6.3.3. Solution via Jordan Form 70 Oxford Physics Lectures: Ordinary Differential Equations 3 7. Normal Modes of Linear Oscillatory Systems 74 7.1. Coupled Identical Oscillators 74 7.1.1. General Method 74 7.1.2. Solution for Two Coupled Pendula 76 7.1.3. Examples of Initial Conditions 77 7.2. Energetics of Coupled Oscillators 78 7.2.1. Energy of a System of Coupled Oscillators 78 7.2.2. Restoring Forces Are Potential 79 7.2.3. Energy in Terms of Normal Modes 79 7.3. Generalisations and Complications 80 7.3.1. Pendula of Unequal Length 80 7.3.2. Pendula of Unequal Mass 80 7.3.3. Damped Oscillators 82 7.3.4. Forced Oscillators and Resonances 83 7.3.5. A Worked Example: Can Coupled Pendula Be in Resonance With Each Other? 85 7.3.6. N Coupled Oscillators 86 7.4. (Nonlinear) Digression: Coupled Oscillators and Hamiltonian Chaos 88 8. Qualitative Solution of Systems of Autonomous Nonlinear ODEs 88 8.1. Classification of 2D Equilibria 90 8.1.1. Nodes: T 2 > 4D > 0 90 8.1.2. Saddles: T 2 > 0 > 4D 92 8.1.3. Pathologies: D = 0 92 8.1.4. Effect of Nonlinearity 93 8.1.5. Pathologies: T 2 = 4D 94 8.1.6. Foci: 4D > T 2 > 0 95 8.1.7. Centres: 4D > T 2 = 0 97 8.1.8. Conservative Systems 97 8.2. (Nonlinear) Digression: Hamiltonian Systems and Adiabatic Invariants 99 8.3. Limit Cycles 102 8.3.1. Poincar´e{Bendixson Theorem 103 8.4. Auto-Oscillations 105 8.4.1. Li´enard'sTheorem 107 8.4.2. Relaxation Oscillations 107 8.5. Outlook: Attractors, Dissipative Chaos, and Many Degrees of Freedom 109 Appendix: Introduction to Lagrangian and Hamiltonian Mechanics 110 A.1. Hamilton's Action Principle 110 A.2. Lagrangian of a Point Mass 111 A.3. Lagrangian of a System of Point Masses 111 A.4. Momentum 112 A.5. Energy 112 A.6. Hamilton's Equations of Motion 113 A.7. Liouville's Theorem 114 Sources and Further Reading 116 Problem Sets i PS-1: First-Order ODEs i PS-2: Second-Order ODEs, Part I vi PS-3: Second-Order ODEs, Part II x PS-4: Systems of Linear ODEs xiii 4 A. A. Schekochihin PS-5: Systems of Nonlinear ODEs xv PS-6: Masses on Springs and Other Things xvii Oxford Physics Lectures: Ordinary Differential Equations 5 To play the good family doctor who warns about reading something prematurely, simply because it would be premature for him his whole life long|I'm not the man for that. And I find nothing more tactless and brutal than constantly trying to nail talented youth down to its \immaturity," with every other sentence a \that's nothing for you yet." Let him be the judge of that! Let him keep an eye out for how he manages. Thomas Mann, Doctor Faustus 1. The Language of the Game It might not be an exaggeration to say that the ability of physical theory to make pre- dictions and, consequently, both what we proudly call our \understanding of the world" and our technological civilisation hinge on our ability to solve differential equations|or, at least, to write them down and make intelligent progress in describing their solutions in qualitative, asymptotic or numerical terms. Thus, what you are about to start learning may well be the most \practically" important bit of mathematics for you as physicists|so be excited! I shall start by introducing some terms and notions that will form the language that we will speak and by stating some general results that enable us to speak it meaningfully. Then I shall move on to methods for solving and analysing various kinds of differential equations. 1.1. What is an ODE? [Literature: Pontryagin(1962, x1), Tikhonov et al. (1985, xx1.1-1.2), Arnold(2006, xx1.1-1.10), Yeomans(2014, xII.0), Tenenbaum & Pollard(1986, x3), Bender & Orszag(1999, x1.1)] A differential equation is an equation in which the unknowns are functions of one or more variables and which contains both these functions and their derivatives. Physically, it is a relationship between quantities and their rates of change|in time, space or whatever other \independent variable" we care to introduce. If more than one independent variable is involved, the differential equation is called a partial differential equation (PDE). Those will not concern me|you will encounter them next term, in CP4, and also, in a proper way, in the second-year Mathematical Methods (Eßler 2009; Magorrian 2017; Lukas 2019). It makes sense that this should happen after you have|hopefully|learned from me how to handle ordinary differential equations (ODEs), which are differential equations that involve functions of a single independent variable. They have the general form F (x; y; y0; : : : ; y(n)) = 0 ; (1.1) where x is the independent variable, y is the function (or \dependent variable"1) and y0, y00,..., y(n) are its first, second, .
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