- 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

# Lagrangian Mechanics

# Lagrangian Mechanics

- Chapter:
- (p.107) 6 Lagrangian Mechanics
- Source:
- The Lazy Universe
- Author(s):
### Jennifer Coopersmith

- Publisher:
- Oxford University Press

It is demonstrated how d’Alembert’s Principle can be used as the basis for a more general mechanics – Lagrangian Mechanics. How this leads to Hamilton’s Principle (the Principle of Least Action) is shown mathematically and in words. It is further explained why Lagrangian Mechanics is so general, why forces of constraint may be ignored, and how external conditions lead to “curved space.” Also, it is explained why the Lagrangian, *L*, has the form *L* = *T* − *V* (where *T* is the kinetic energy and *V* is the potential energy), and why *T* is in “quadratic form” (*T* = 1/2*mv*^{2}). It is shown how Noether’s Theorem leads to a more fundamental definition of energy and links the conservation of energy to the homogeneity of time. The ingenious Lagrange multipliers are explained, and also generalized forces and generalized coordinates.

*Keywords:*
L = T − V, T = 1/2mv2, quadratic form, Noether’s Theorem, Lagrange multipliers, generalized forces, generalized coordinates, curved space, Hamilton’s Principle

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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