MATH JOURNEY THROUGH DIFFERENTIAL EQUATIONS OF PHYSICS

Mathematics is the language of physics, and over time physicists have developed their own dialect. The main purpose of this book is to bridge this language barrier, and introduce the readers to the beauty of mathematical physics. It shows how to combine the strengths of both approaches: physicists often arrive at interesting conjectures based on good intuition, which can serve as the starting point of interesting mathematics. Conversely, mathematicians can more easily see commonalities between very different fields (such as quantum mechanics and electromagnetism), and employ more advanced tools. Rather than focusing on a particular topic, the book showcases conceptual and mathematical commonalities across different physical theories. It translates physical problems to concrete mathematical questions, shows how to answer them and explains how to interpret the answers physically. For example, if two Hamiltonians are close, why are their dynamics similar? The book alternates between mathematics- and physics-centric chapters, and includes plenty of concrete examples from physics as well as 76 exercises with solutions. It exploits that readers from either end are familiar with some of the material already. The mathematics-centric chapters provide the necessary background to make physical concepts mathematically precise and establish basic facts. And each physics-centric chapter introduces physical theories in a way that is more friendly to mathematicians. As the book progresses, advanced material is sprinkled in to showcase how mathematics and physics augment one another. Some of these examples are based on recent publications and include material which has not been covered in other textbooks. This is to keep it interesting for the readers. Contents: Preface About the Author Acknowledgments Introduction Ordinary Differential Equations The Hamiltonian Formalism of Classical Mechanics Banach & Hilbert Spaces Linear Operators The Fourier Transform Schwartz Functions & Tempered Distributions Green's Functions Quantum Mechanics Variational Calculus Appendix A: Primer on Measure Theory Appendix B: Functional Calculus Solutions to the Odd-Numbered Exercises Bibliography Index Readership: Advanced undergraduate and graduate students of mathematics and physics with an interest in mathematical physics. Instructors can use it to design a comprehensive course on differential equations after paring down some of the material. Or alternatively, it also serves as a good basis for more specialized classes on, e.g., quantum mechanics or electromagnetism that place more emphasis on the necessary mathematics. Key Features: This book is aimed to appeal to physicists and mathematicians alike. Its focus is not one particular theory, but rather emphasizes mathematical and physical similarities and differences between different physical systems and the differential equations describing them This book covers beyond standard textbook fare where the chapters are enriched with research-level material The purpose is to illustrate how to solve concrete problems from physics with the mathematical tools developed up until that point The materials in the book are designed to be optional, thus instructors can pick and choose without having to change the flow of their course The book was developed from lecture notes of a 2-semester course for third- and fourth-year students. Material covered in this book is relevant to courses in ordinary differential equations, classical mechanics, operator theory, partial differential equations, quantum mechanics and the calculus of variations This book summarizes many of the standard facts on differential equations, and could be used as an encyclopaedia for the most important ideas and techniques Curated references will point interested readers to the right direction should they want to know more about any particular subject