Introduction to Methods of Approximation in Physics and Astronomy

347 Pages·2017·6.68 MB·English

by  Maurice H. P. M. van Putten

Undergraduate Lecture Notes in Physics
Maurice H.P.M. van Putten
Introduction
to Methods of
Approximation
in Physics and
Astronomy
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Maurice H.P.M. van Putten
Introduction to Methods
of Approximation in Physics
and Astronomy
123
Maurice H.P.M.vanPutten
Department ofPhysics andAstronomy
SejongUniversity
Seoul
Republicof Korea (SouthKorea)
ISSN 2192-4791 ISSN 2192-4805 (electronic)
Undergraduate Lecture Notesin Physics
ISBN978-981-10-2931-8 ISBN978-981-10-2932-5 (eBook)
DOI 10.1007/978-981-10-2932-5
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To my parents
Preface
ModernastronomyrevealsanevolvingUniverserifewithtransientsources,mostly
discovered—few predicted—in multi-wavelength observations. Our window of
observations now includes electromagnetic radiation, gravitational waves and
neutrinos. For the practicing astronomer, these are highly interdisciplinary devel-
opments that pose a novel challenge to be well-versed in astroparticle physics and
data analysis. In realizing the full discovery potential of these multimessenger
approaches, the latter increasingly involves high-performance supercomputing.
These lecture notes developed out of lectures on mathematical-physics in
astronomy to advanced undergraduate and beginning graduate students. They are
organizedtobe largely self-contained, starting from basic conceptsand techniques
in the formulation of problems and methods of approximation commonly used in
computation and numerical analysis. This includes root finding, integration, signal
detection algorithms involving the Fourier transform and examples of numerical
integration of ordinary differential equations and some illustrative aspects of
modern computational implementation. In the applications, considerable emphasis
is put on fluid dynamical problems associated with accretion flows, as these are
responsible for a wealth of high energy emission phenomena in astronomy.
Thetopicschosenarelargelyaimedatphenomenologicalapproaches,tocapture
mainfeaturesofinterestbyeffectivemethodsofapproximationatadesiredlevelof
accuracy and resolution. Formulated in terms of a system of algebraic, ordinary or
partial differential equations, this may be pursued by perturbation theory through
expansions in a small parameter or by direct numerical computation. Successful
application of these methods requires a robust understanding of asymptotic
behavior,errorsandconvergence.Insomecases,thenumberofdegreesoffreedom
may be reduced, e.g., for the purpose of (numerical) continuation or to identify
secular behavior. For instance, secular evolution of orbital parameters may derive
from averaging over essentially periodic behavior on relatively short, orbital peri-
ods. When the original number of degrees of freedom is large, averaging over
dynamical time scales may lead to a formulation in terms of a system in approxi-
matelythermodynamicequilibriumsubjecttoevolutiononaseculartimescalebya
regular or singular perturbation.
vii
viii Preface
In modern astrophysics and cosmology, gravitation is being probed across an
increasingly broad range of scales and more accurately so than ever before. These
observations probe weak gravitational interactions below what is encountered in
our solar system by many orders of magnitude. These observations hereby probe
(curved) spacetime at low energy scales that may reveal novel properties hitherto
unanticipated in the classical vacuum of Newtonian mechanics and Minkowski
spacetime. Dark energy and dark matter encountered on the scales of galaxies and
beyond, therefore, may be, in part, revealing our ignorance of the vacuum at the
lowest energy scales encountered in cosmology. In this context, our application of
Newtonian mechanics to globular clusters, galaxies and cosmology is an approxi-
mation assuming a classical vacuum, ignoring the potential for hidden low energy
scales emerging on cosmological scales. Given our ignorance of the latter, this
posesachallengeinthepotentialforunknownsystematicdeviations.Ifofquantum
mechanical origin, such deviations are often referred to as anomalies. While they
aresmallintraditional,macroscopicNewtonianexperimentsinthelaboratory,they
same is not a given in the limit of arbitrarily weak gravitational interactions.
Wehopethisselectionofintroductorymaterialisusefulandkindlesthereader’s
interest to become a creative member of modern astrophysics and cosmology.
Thisbookwouldnothavebeenpossiblewithoutnumerousin-classinteractions
that largely shaped the choice of materials and method of presentation. The author
gratefullythanksthemanystudentswhoparticipatedinthisdevelopment.Someof
this work is supported by the National Research Foundation of Korea under Grant
Nos. 2015R1D1A1A01059793 and 2016R1A5A1013277.
Seoul, Republic of Korea (South Korea) Maurice H.P.M. van Putten
Contents
Part I Preliminaries
1 Complex Numbers ... .... ..... .... .... .... .... .... ..... .. 3
1.1 Quotients of Complex Numbers.. .... .... .... .... ..... .. 7
1.2 Roots of Complex Numbers. .... .... .... .... .... ..... .. 8
1.3 Sequences and Euler’s Constant.. .... .... .... .... ..... .. 9
1.4 Power Series and Radius of Convergence .. .... .... ..... .. 13
1.5 Minkowski Spacetime ..... .... .... .... .... .... ..... .. 17
1.5.1 Rindler Observers.. .... .... .... .... .... ..... .. 20
1.6 The Logarithm and Winding Number . .... .... .... ..... .. 24
1.7 Branch Cuts for logZ. ..... .... .... .... .... .... ..... .. 25
1
1.8 Branch Cuts for zp ... ..... .... .... .... .... .... ..... .. 27
1.9 Exercises .. .... .... ..... .... .... .... .... .... ..... .. 29
Reference ... .... .... .... ..... .... .... .... .... .... ..... .. 33
2 Complex Function Theory. ..... .... .... .... .... .... ..... .. 35
2.1 Analytic Functions... ..... .... .... .... .... .... ..... .. 35
2.2 Cauchy’s Integral Formula.. .... .... .... .... .... ..... .. 38
2.3 Evaluation of a Real Integral.... .... .... .... .... ..... .. 42
2.4 Residue Theorem.... ..... .... .... .... .... .... ..... .. 43
2.5 Morera’s Theorem ... ..... .... .... .... .... .... ..... .. 48
2.6 Liouville’s Theorem.. ..... .... .... .... .... .... ..... .. 50
2.7 Poisson Kernel.. .... ..... .... .... .... .... .... ..... .. 51
2.8 Flux and Circulation . ..... .... .... .... .... .... ..... .. 52
2.9 Examples of Potential Flows .... .... .... .... .... ..... .. 55
2.10 Exercises .. .... .... ..... .... .... .... .... .... ..... .. 56
References .. .... .... .... ..... .... .... .... .... .... ..... .. 60
ix
x Contents
3 Vectors and Linear Algebra..... .... .... .... .... .... ..... .. 61
3.1 Introduction .... .... ..... .... .... .... .... .... ..... .. 61
3.2 Inner and Outer Products... .... .... .... .... .... ..... .. 63
3.3 Angular Momentum Vector. .... .... .... .... .... ..... .. 63
3.3.1 Rotations.... ..... .... .... .... .... .... ..... .. 64
3.3.2 Angular Momentum and Mach’s Principle ... ..... .. 65
3.3.3 Energy and Torque. .... .... .... .... .... ..... .. 66
3.3.4 Coriolis Forces .... .... .... .... .... .... ..... .. 70
3.3.5 Spinning Top ..... .... .... .... .... .... ..... .. 71
3.4 Elementary Transformations in the Plane... .... .... ..... .. 74
3.4.1 Reflection Matrix .. .... .... .... .... .... ..... .. 74
3.4.2 Rotation Matrix.... .... .... .... .... .... ..... .. 76
3.5 Matrix Algebra.. .... ..... .... .... .... .... .... ..... .. 77
3.6 Eigenvalue Problems . ..... .... .... .... .... .... ..... .. 78
3.6.1 Eigenvalues of Rð’Þ.... .... .... .... .... ..... .. 78
3.6.2 Eigenvalues of a Real-Symmetric Matrix .... ..... .. 79
3.6.3 Hermitian Matrices. .... .... .... .... .... ..... .. 80
3.7 Unitary Matrices and Invariants.. .... .... .... .... ..... .. 83
3.8 Hermitian Structure of Minkowski Spacetime ... .... ..... .. 86
3.9 Eigenvectors of Hermitian Matrices... .... .... .... ..... .. 91
3.10 QR Factorization .... ..... .... .... .... .... .... ..... .. 94
3.10.1 Examples of Image and Null Space .... .... ..... .. 96
3.10.2 Dimensions of Image and Null Space... .... ..... .. 97
3.10.3 QR Factorization by Gram-Schmidt .... .... ..... .. 100
3.11 Exercises .. .... .... ..... .... .... .... .... .... ..... .. 101
References .. .... .... .... ..... .... .... .... .... .... ..... .. 107
4 Linear Partial Differential Equations . .... .... .... .... ..... .. 109
4.1 Hyperbolic Equations. ..... .... .... .... .... .... ..... .. 109
4.1.1 Inhomogeneous Wave Equation (Duhamel) .. ..... .. 113
4.2 Diffusion Equation... ..... .... .... .... .... .... ..... .. 115
4.2.1 Photon Diffusion in the Sun .. .... .... .... ..... .. 121
4.3 Elliptic Equations.... ..... .... .... .... .... .... ..... .. 122
4.4 Characteristics of Hyperbolic Systems. .... .... .... ..... .. 125
4.5 Weyl Equation.. .... ..... .... .... .... .... .... ..... .. 126
4.6 Exercises .. .... .... ..... .... .... .... .... .... ..... .. 129
References .. .... .... .... ..... .... .... .... .... .... ..... .. 131
Part II Methods of Approximation
5 Projections and Minimal Distances ... .... .... .... .... ..... .. 135
5.1 Vectors and Distances ..... .... .... .... .... .... ..... .. 135
5.2 Projections of Vectors ..... .... .... .... .... .... ..... .. 137
5.3 Snell’s Law and Fermat’s Principle ... .... .... .... ..... .. 142