Analytical Mechanics for Relativity And Quantum Mechanics
This book provides 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 graduate level.A distinguishing feature of the book is its integration of special relativity into the teaching of classical mechanics. Extended Lagrangian and Hamiltonian methods are introduced that treat time as a transformable coordinate rather than the fixed parameter of Newtonian physics. Advanced topics such as covariant Lagrangians and Hamiltonians, canonical transformations, and the Hamilton-Jacobi equation are developed using this extended theory. This permits the Lorentz transformation of special relativity to become a canonical transformation.This is also a book for those who study analytical mechanics as a preliminary to a critical exploration of quantum mechanics. Comparisons to quantum mechanics appear throughout the text, and classical mechanics itself is presented in a way that will aid the reader in the study of quantum theory. A chapter is devoted to linear vector operators and dyadics, including a comparison to the bra-ket notation of quantum mechanics. Rotations are presented using an operator formalism similar to that used in quantum theory, and the definition of the Euler angles follows the quantum mechanical convention. 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.The book provides a necessary bridge to carry graduate students from their previous undergraduate classical mechanics courses to the future study of advanced relativity and 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. This text is intended to encourage the retention or restoration of introductory graduate analytical mechanics courses. It is written for the intellectually curious graduate student, and the teacher who values mathematical precision in addition to accessibility.
Part I: Introduction: The Traditional Theory1. Basic Dynamics of Point Particles and Collections2. Introduction to Lagrangian Mechanics3. Lagrangian Theory of Constraints4. Introduction to Hamiltonian Mechanics5. The Calculus of Variations6. Hamilton's Principle7. Linear Operators and Dyadics8. Kinematics of Rotation9. Rotational Dynamics10. Small Vibrations about EquilibriumPart II: Mechanics with Time as a Coordinate11. Lagrangian Mechanics with Time as a Coordinate12. Hamiltonian Mechanics with Time as a Coordinate13. Hamilton's Principle and Noether's Theorem14. Relativity and Spacetime15. Fourvectors and Operators16. Relativistic Mechanics17. Canonical Transformations18. Generating Functions19. Hamilton-Jacobi TheoryPart III: Mathematical AppendicesA. Vector FundamentalsB. Matrices and DeterminantsC. Eigenvalue Problem with General MetricD. The Calculus of Many VariablesE. Geometry of Phase Space