- Title Pages
- Dedication
- Preface to Second Edition
- Preface to First Edition
- Acknowledgments
- 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
- 11 Central Force Motion
- 12 Scattering
- 13 Lagrangian Mechanics with Time as a Coordinate
- 14 Hamiltonian Mechanics with Time as a Coordinate
- 15 Hamilton'S Principle and Noether's Theorem
- 16 Relativity and Spacetime
- 17 Fourvectors and Operators
- 18 Relativistic Mechanics
- 19 Canonical Transformations
- 20 Generating Functions
- 21 Hamilton-Jacobi Therory
- 22 Angle‐Action Variables
- Appendix A Vector Fundamentals
- Appendix B Matrices and Determinants
- Appendix C Eigenvalue Problem with General Metric
- Appendix D The Calculus of Many Variables
- Appendix E Geometry of Phase Space
- References
- Index
Introduction to Hamiltonian Mechanics
Introduction to Hamiltonian Mechanics
- Chapter:
- (p.70) 4 Introduction to Hamiltonian Mechanics
- Source:
- Analytical Mechanics for Relativity and Quantum Mechanics
- Author(s):
Oliver Davis Johns
- Publisher:
- Oxford University Press
This chapter argues that Hamiltonian mechanics is a much better base from which to build more advanced methods. The Hamilton equations have an elegant symmetry that the Lagrange equations lack. The Hamiltonian function is also used to write the Schroedinger equation of quantum mechanics. The differences between the Lagrange and Hamilton equations result mainly from the different variable sets in which they act. The Lagrangian variable set is the set of generalised coordinates and velocities, whereas the Hamiltonian set is the set of generalised coordinates and momenta. The transformation from Lagrange to Hamilton equations is a Legendre transformation, in which the Lagrangian function of the Lagrangian variable set is to be replaced by the Hamiltonian function of the phase-space variable set.
Keywords: quantum mechanics, Hamiltonian mechanics, Hamilton equations, Lagrange equations, Schroedinger equation
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- Title Pages
- Dedication
- Preface to Second Edition
- Preface to First Edition
- Acknowledgments
- 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
- 11 Central Force Motion
- 12 Scattering
- 13 Lagrangian Mechanics with Time as a Coordinate
- 14 Hamiltonian Mechanics with Time as a Coordinate
- 15 Hamilton'S Principle and Noether's Theorem
- 16 Relativity and Spacetime
- 17 Fourvectors and Operators
- 18 Relativistic Mechanics
- 19 Canonical Transformations
- 20 Generating Functions
- 21 Hamilton-Jacobi Therory
- 22 Angle‐Action Variables
- Appendix A Vector Fundamentals
- Appendix B Matrices and Determinants
- Appendix C Eigenvalue Problem with General Metric
- Appendix D The Calculus of Many Variables
- Appendix E Geometry of Phase Space
- References
- Index
