- 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
Lagrangian Theory of Constraints
Lagrangian Theory of Constraints
- Chapter:
- (p.46) 3 Lagrangian Theory of Constraints
- Source:
- Analytical Mechanics for Relativity and Quantum Mechanics
- Author(s):
Oliver Davis Johns
- Publisher:
- Oxford University Press
This chapter discusses how easy the Lagrangian method solves so-called constraint problems. Presented here are different methods of solving such problems, with corresponding examples. Constraints can be incorporated into the Lagrangian method in a particularly convenient way; if the constraints are idealised – such as frictionless surfaces or perfectly rigid bodies – then the equations of motion can be solved without knowing the forces of constraint. Also, the number of degrees of freedom of the Lagrangian system can be reduced by one for each constraint applied. The chapter also defines constraints, beginning with holonomic ones, which are the simplest class of constraints. A constraint is holonomic if it can be represented by a single function of the generalised coordinates, equated to zero.
Keywords: constraint problems, Lagrangian method, frictionless surfaces, perfectly rigid bodies, forces of constraint
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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
