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Action and the Principle of Least Action are explained: what Action is, why the Principle of Least Action works, why it underlies all physics, and what are the insights gained into energy, space, and time. The physical and mathematical origins of the Lagrange Equations, Hamilton’s Equations, the Lagrangian, the Hamiltonian, and the Hamilton-Jacobi Equation are shown. Also, worked examples in Lagrangian and Hamiltonian Mechanics are given. However the aim is to explain physics rather than to give a technical mastery of the subject. Therefore, much of the mathematics is in the appendices. While ... More

*Keywords: *
Least Action,
Lagrangian Mechanics,
Hamiltonian Mechanics,
Variational Mechanics,
Cornelius Lanczos,
Hamilton-Jacobi Equation,
phase space,
Fermat’s Principle,
Noether’s Theorem

Print publication date: 2017 | Print ISBN-13: 9780198743040 |

Published to Oxford Scholarship Online: June 2017 | DOI:10.1093/oso/9780198743040.001.0001 |

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## Front Matter

## End Matter

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Appendix A1.1 Newton’s Laws of Motion

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Appendix A2.1 Portraits of the physicists

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Appendix A3.1 Reversible displacements

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Appendix A6.1 Worked examples in Lagrangian Mechanics

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Appendix A6.2 Proof that T is a function of v2

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Appendix A6.3 Energy conservation and the homogeneity of time

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Appendix A6.4 The method of Lagrange Multipliers

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Appendix A6.5 Generalized Forces

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Appendix A7.1 Hamilton’s Transformation, examples

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Appendix A7.2 Demonstration that the pi s are independent coordinates

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Appendix A7.3 Worked examples in Hamiltonian Mechanics

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Appendix A7.4 Incompressibility of the phase fluid

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Appendix A7.5 Energy conservation in extended phase space

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Appendix A7.6 Link between the action, S, and the ‘circulation’

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Appendix A7.7 Transformation equations linking p and q via S

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Appendix A7.8 Infinitesimal canonical transformations

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Appendix A7.9 Perpendicularity of wavefronts and rays

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Appendix A7.10 Problems solved using the Hamilton-Jacobi Equation

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Appendix A7.11 Quasi refractive index in mechanics

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Appendix A7.12 Einstein’s link between Action and the de Broglie waves

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Bibliography and Further Reading

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Index

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