 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 HamiltonJacobi 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
Hamiltonian Mechanics with Time as a Coordinate
Hamiltonian Mechanics with Time as a Coordinate
 Chapter:
 (p.308) 14 Hamiltonian Mechanics with Time as a Coordinate
 Source:
 Analytical Mechanics for Relativity and Quantum Mechanics
 Author(s):
Oliver Davis Johns
 Publisher:
 Oxford University Press
This chapter uses the traditional Hamilton equations as the basis for an extended Hamiltonian theory in which time is treated as a coordinate. The traditional Hamilton equations, including the Hamiltonian form of the generalised energy theorem, will be combined into one set of extended Hamilton equations. The extended Hamilton theory developed in the chapter is of fundamental importance for the more advanced topics in mechanics. It is used to write the relativistically covariant Hamiltonian, which is then used to derive the KleinGordon equation of relativistic quantum mechanics. The extended Hamilton equations also provide the basis for the discussion of canonical transformations. The objective of extended Hamiltonian theory is to write the equations of motion in terms of an extended set of phasespace variables.
Keywords: traditional Hamilton equations, extended Hamiltonian theory, generalised energy theorem, relativistically covariant Hamiltonian, KleinGordon equation, relativistic quantum mechanics
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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 HamiltonJacobi 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