Analytical Mechanics for Relativity and Quantum Mechanics

Description

Analytical Mechanics for Relativity and Quantum Mechanics is an innovative and mathematically sound treatment of the foundations of analytical mechanics and the relation of classical mechanics to relativity and quantum theory. It is intended for use at the introductory graduate level. A distinguishing feature of the book is its integration of special relativity into teaching of classical mechanics. After a thorough review of the traditional theory, Part II of the book introduces extended Lagrangian and Hamiltonian methods that treat time as a transformable coordinate rather than the fixed parameter of Newtonian physics. Advanced topics such as covariant Langrangians and Hamiltonians, canonical transformations, and Hamilton-Jacobi methods are simplified by the use of this extended theory. And the definition of canonical transformation no longer excludes the Lorenz transformation of special relativity. This is also a book for those who study analytical mechanics to prepare for a critical exploration of quantum mechanics. Comparisons to quantum mechanics appear throughout the text. The extended Hamiltonian theory with time as a coordinate is compared to Dirac's formalism of primary phase space constraints. The chapter on relativistic mechanics shows how to use covariant Hamiltonian theory to write the Klein-Gordon and Dirac equations. The chapter on Hamilton-Jacobi theory includes a discussion of the closely related Bohm hidden variable model of quantum mechanics. Classical mechanics itself is presented with an emphasis on methods, such as linear vector operators and dyadics, that will familiarize the student with similar techniques in quantum theory. Several of the current fundamental problems in theoretical physics - the development of quantum information technology, and the problem of quantizing the gravitational field, to name two - require a rethinking of the quantum-classical connection. Graduate students preparing for research careers will find a graduate mechanics course based on this book to be an essential bridge between their undergraduate training and advanced study in analytical mechanics, relativity, and quantum mechanics Content: PART I: THE CLASSICAL THEORY 1. Basic Dynamics of Point Particles and Collections 2. Introduction to Lagrangian Mechanics 3. Lagrangian Theory of Constraints 4. Introduction to Hamiltonian Mechanics 5. The Calculus of Variations 6. Hamilton's Principle 7. Linear Operators and Dyadics 8. Kinematics of Rotation 9. Rotational Dynamics 10. Small Vibrations about Equilibrium PART II: MECHANICS WITH TIME AS A COORDINATE 11. Lagrangian Mechanics with Time as a Coordinate 12. Hamiltonian Mechanics with Time as a Coordinate 13. Hamilton's Principle and Noether's Theorem 14. Relativity and Spacetime 15. Fourvectors and Operators 16. Relativistic Mechanics 17. Canonical Transformations with Time as a Coordinate 18. Generating Functions 19. Hamilton-Jacobi Theory PART III: MATHEMATICAL REFERENCES A. Vector Fundamentals B. Matrices and Determinants C. Eigenvalue Problem with General Metric D. The Calculus of Many Variables E. Geometry of Phase Space