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An introductory textbook exploring the subject of Lagrangian and Hamiltonian dynamics, with a relaxed and self-contained setting. Lagrangian and Hamiltonian dynamics is the continuation of Newton's classical physics into new formalisms, each highlighting novel aspects of mechanics that gradually build in complexity to form the basis for almost all of theoretical physics. Lagrangian and Hamiltonian dynamics also acts as a gateway to more abstract concepts routed in
differential geometry and field theories and can be used to introduce these subject areas to newcomers.
Journeying in a self-contained manner from the very basics, through the fundamentals and onwards to the cutting edge of the subject, along the way the reader is supported by all the necessary background mathematics, fully worked examples, thoughtful and vibrant illustrations as well as an informal narrative and numerous fresh, modern and inter-disciplinary applications.
The book contains some unusual topics for a classical mechanics textbook. Most notable examples include the 'classical wavefunction', Koopman-von Neumann theory, classical density functional theories, the 'vakonomic' variational principle for non-holonomic constraints, the Gibbs-Appell equations, classical path integrals, Nambu brackets and the full framing of mechanics in the language of differential geometry.
List of contents
- Part I: Newtonian Mechanics
- 1: Introduction
- 2: Newton's Three Laws
- 3: Energy and Work
- 4: Introductory Rotational Dynamics
- 5: The Harmonic Oscillator
- 6: Wave Mechanics and Elements of Mathematical Physics
- Part II: Langrangian Mechanics
- 7: Introduction
- 8: Coordinates and Constraints
- 9: The Stationary Action Principle
- 10: Constrained Langrangian Mechanics
- 11: Point Transformations in Langrangian Mechanics
- 12: The Jacobi Energy Function
- 13: Symmetries and Langrangian-Hamiltonian-Jacobi Theory
- 14: Near-Equilibrium Oscillations
- 15: Virtual Work and d'Alembert's Principle
- Part III: Canonical Mechanics
- 16: Introduction
- 17: The Hamiltonian and Phase Space
- 18: Hamiltonian's equations and Routhian Reduction
- 19: Poisson Brackets and Angular momentum
- 20: Canonical and Gauge Transformations
- 21: Hamilton-Jacobi Theory
- 22: Liouville's Theorem and Classical Statistical Mechanics
- 23: Constrained Hamiltonian Dynamics
- 24: Autonomous Geometrical Mehcanics
- 25: The Structure of Phase Space
- 26: Near-Integrable Systems
- Part IV: Classical Field Theory
- 27: Introduction
- 28: Langrangian Field Theory
- 29: Hamiltonian Field Theory
- 30: Clssical Electromagnetism
- 31: Neother's Theorem for Fields
- 32: Classical Path-Integrals
- Part V: Preliminary Mathematics
- 33: The (Not so?) Basics
- 34: Matrices
- 35: Partial Differentiation
- 36: Legendre Transformations
- 37: Vector Calculus
- 38: Differential equations
- 39: Calculus of Variations
- Part VI: Advanced Mathematics
- 40: Linear Algebra
- 41: Differential Geometry
- Part VII: Exam Style Questions
- Appendix A: Noether's Theorem Explored
- Appendix B: The Action Principle Explored
- Appendix C: Useful Relations
- Appendxi D: Poisson and Nambu Brackets Explored
- Appendix: Canonical Transformations Explored
- Appendix F: Action-Angle Variables Explored
- Appendix G: Statistical Mechanics Explored
- Appendix H: Biographies
About the author
Peter Mann completed his undergraduate degree in Chemistry at the University of St Andrews. He is now a PhD student at the University of St Andrews investigating spreading phenomena on complex networks and how antibiotic resistance proliferates on different network topologies.
Summary
The book introduces classical mechanics. It does so in an informal style with numerous fresh, modern and inter-disciplinary applications assuming no prior knowledge of the necessary mathematics. The book provides a comprehensive and self-contained treatment of the subject matter up to the forefront of research in multiple areas.
