- Title Pages
- Frontispiece
- Dedication
- Epigraph
- Preface
- List of Figures
- 1 Introduction
- 2 Antecedents
- 3 Mathematics and physics preliminaries: of hills and plains and other things
- 4 The Principle of Virtual Work
- 5 D’Alembert’s Principle
- 6 Lagrangian Mechanics
- 7 Hamiltonian Mechanics
- 8 The whole of physics
- 9 Final words
- Appendix A1.1 Newton’s Laws of Motion
- Appendix A2.1 Portraits of the physicists
- Appendix A3.1 Reversible displacements
- Appendix A6.1 Worked examples in Lagrangian Mechanics
- Appendix A6.2 Proof that <i>T</i> is a function of <i>v</i><sup>2</sup>
- Appendix A6.3 Energy conservation and the homogeneity of time
- Appendix A6.4 The method of Lagrange Multipliers
- Appendix A6.5 Generalized Forces
- Appendix A7.1 Hamilton’s Transformation, examples
- Appendix A7.2 Demonstration that the pi s are independent coordinates
- Appendix A7.3 Worked examples in Hamiltonian Mechanics
- Appendix A7.4 Incompressibility of the phase fluid
- Appendix A7.5 Energy conservation in extended phase space
- Appendix A7.6 Link between the action, <i>S</i>, and the ‘circulation’
- Appendix A7.7 Transformation equations linking <i>p</i> and <i>q</i> via <i>S</i>
- Appendix A7.8 Infinitesimal canonical transformations
- Appendix A7.9 Perpendicularity of wavefronts and rays
- Appendix A7.10 Problems solved using the Hamilton-Jacobi Equation
- Appendix A7.11 Quasi refractive index in mechanics
- Appendix A7.12 Einstein’s link between Action and the de Broglie waves
- Bibliography and Further Reading
- Index

# Hamiltonian Mechanics

# Hamiltonian Mechanics

- Chapter:
- (p.142) (p.143) 7 Hamiltonian Mechanics
- Source:
- The Lazy Universe
- Author(s):
### Jennifer Coopersmith

- Publisher:
- Oxford University Press

Hamilton’s genius was to understand what were the true variables of mechanics (the “*p* − *q*,” conjugate coordinates, or canonical variables), and this led to Hamilton’s Mechanics which could obtain qualitative answers to a wider ranger of problems than Lagrangian Mechanics. It is explained how Hamilton’s canonical equations arise, why the Hamiltonian is the “central conception of all modern theory” (quote of Schrödinger’s), what the “*p* − *q*” variables are, and what phase space is. It is also explained how the famous conservation theorems arise (for energy, linear momentum, and angular momentum), and the connection with symmetry. The Hamilton-Jacobi Equation is derived using infinitesimal canonical transformations (ICTs), and predicts wavefronts of “common action” spreading out in (configuration) space. An analogy can be made with geometrical optics and Huygen’s Principle for the spreading out of light waves. It is shown how Hamilton’s Mechanics can lead into quantum mechanics.

*Keywords:*
Hamiltonian, phase space, Hamilton-Jacobi, conjugate coordinates, canonical equations, infinitesimal canonical transformations, geometrical optics, Huygen’s Principle, Schrödinger, symmetry

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- Title Pages
- Frontispiece
- Dedication
- Epigraph
- Preface
- List of Figures
- 1 Introduction
- 2 Antecedents
- 3 Mathematics and physics preliminaries: of hills and plains and other things
- 4 The Principle of Virtual Work
- 5 D’Alembert’s Principle
- 6 Lagrangian Mechanics
- 7 Hamiltonian Mechanics
- 8 The whole of physics
- 9 Final words
- Appendix A1.1 Newton’s Laws of Motion
- Appendix A2.1 Portraits of the physicists
- Appendix A3.1 Reversible displacements
- Appendix A6.1 Worked examples in Lagrangian Mechanics
- Appendix A6.2 Proof that <i>T</i> is a function of <i>v</i><sup>2</sup>
- Appendix A6.3 Energy conservation and the homogeneity of time
- Appendix A6.4 The method of Lagrange Multipliers
- Appendix A6.5 Generalized Forces
- Appendix A7.1 Hamilton’s Transformation, examples
- Appendix A7.2 Demonstration that the pi s are independent coordinates
- Appendix A7.3 Worked examples in Hamiltonian Mechanics
- Appendix A7.4 Incompressibility of the phase fluid
- Appendix A7.5 Energy conservation in extended phase space
- Appendix A7.6 Link between the action, <i>S</i>, and the ‘circulation’
- Appendix A7.7 Transformation equations linking <i>p</i> and <i>q</i> via <i>S</i>
- Appendix A7.8 Infinitesimal canonical transformations
- Appendix A7.9 Perpendicularity of wavefronts and rays
- Appendix A7.10 Problems solved using the Hamilton-Jacobi Equation
- Appendix A7.11 Quasi refractive index in mechanics
- Appendix A7.12 Einstein’s link between Action and the de Broglie waves
- Bibliography and Further Reading
- Index