This book is an English translation from a Hungarian book designed for graduate and postgraduate students about the use of variational principles in theoretical physics. Unlike many academic textbooks, it dashes across several lecture disciplines taught in physics courses. It emphasizes and demonstrates the use of the variational technique and philosophy behind the basic laws in mechanics, relativity theory, electromagnetism, and quantum mechanics. The book is meant for advanced students and young researchers in theoretical physics but, also, more experienced researchers can benefit from its reading.
Author(s): Tamás Sándor Biró
Series: SpringerBriefs in Physics
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 112
City: Cham, Switzerland
Tags: Variational Calculus, Action Principle, Virtual Work, Einstein-Hilbert Action
Preface
Acknowledgements
Contents
1 Introduction
1.1 Brief History of Roots
1.1.1 Antiquity: Paradoxes
1.1.2 Medieval Thinking: Transition
1.1.3 Renaissance: Dynamics
1.1.4 Enlightenment: The Global View
1.1.5 Romanticism: Grand Unification
1.1.6 Modern Times: Uncertainty
1.1.7 Contemporary Adventure
1.2 Variational Calculus Basics
1.2.1 Function and Functional
1.2.2 Variation
1.2.3 Higher Functional Derivative
1.3 A Simple Exercise
1.3.1 Completion to a Full Square
1.3.2 Limit of an Inequality
1.3.3 Extremum of a Function
1.3.4 Variation of a Functional
1.4 A Somewhat More Involved Exercise
2 Mechanics: Geometry of Orbits
2.1 The Principle of Virtual Work
2.1.1 Halt on the Slope …
2.1.2 Bascule in Equilibrium
2.2 D'Alambert Principle
2.2.1 Free Circular Motion
2.2.2 Pendulum
2.3 The Action Principle
2.4 Second Variation of the Free Motion
2.5 Gauss Principle
2.6 The Method of Lagrange Multipliers
2.7 Mapertuis Principle and Geodetic Motion
2.8 Legendre Transformation and Action—Angle Variables
2.9 Fermat Principle
3 Gravity: The Optimal Curvature
3.1 Mapertuis Principle in Spacetime
3.2 Motion of Charges, Lorentz Force
3.3 The Weak (Newtonian) Gravitational Field
3.4 Geodesic Motion in General Gravitational Field
3.5 Einstein-Hilbert Action
4 Electrodynamics: Forces, Fields, Waves
4.1 Electrostatic Gauss Principle
4.2 Magnetostatics
4.3 Variational Principle of Electrodynamics
4.4 Electric—Magnetic Duality
4.5 Electromagnetic Waves
4.6 Variational Principle with Gauge Fixing
4.7 Spin One
4.8 Quaternion Formalism
5 Quantum Mechanics: The Most Classical Non-classical Theory
5.1 Stationary Case
5.2 Dynamical Case
5.3 Relativistic Case: Time-Independent Potential
5.4 Relativistic Case: Electromagnetic Potential
5.5 Speculations
Appendix Further Readings
Glossary of People Appearing in the Hungarian Book
