Book contents
- Frontmatter
- Contents
- Preface
- 1 NEWTONIAN GRAVITATION
- 2 MINKOWSKI SPACE–TIME
- 3 THE RELATIVISTIC FORM OF PHYSICAL LAWS
- 4 GRAVITATION AND SPECIAL RELATIVITY
- 5 ELECTROMAGNETISM AND RELATIVISTIC HYDRODYNAMICS
- 6 WHAT IS CURVED SPACE?
- 7 THE PRINCIPLE OF EQUIVALENCE
- 8 EINSTEIN'S RELATIVISTIC GRAVITATION (GENERAL RELATIVITY)
- Appendix A Tensors
- Appendix B Exterior Differential Forms
- Appendix C Variational Form of the Field Equations
- Appendix D The Concept of a Manifold
- References
- Physical Constants
Appendix C - Variational Form of the Field Equations
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 NEWTONIAN GRAVITATION
- 2 MINKOWSKI SPACE–TIME
- 3 THE RELATIVISTIC FORM OF PHYSICAL LAWS
- 4 GRAVITATION AND SPECIAL RELATIVITY
- 5 ELECTROMAGNETISM AND RELATIVISTIC HYDRODYNAMICS
- 6 WHAT IS CURVED SPACE?
- 7 THE PRINCIPLE OF EQUIVALENCE
- 8 EINSTEIN'S RELATIVISTIC GRAVITATION (GENERAL RELATIVITY)
- Appendix A Tensors
- Appendix B Exterior Differential Forms
- Appendix C Variational Form of the Field Equations
- Appendix D The Concept of a Manifold
- References
- Physical Constants
Summary
The usual equations of motion of a particle or a system of particles can be deduced from a variational principle [principle of least action leading either to Lagrange's or Hamilton's equations, depending on the variables used, see H. Goldstein (1980) or L. Landau and E. Lifschitz (1960)], in both the Newtonian and relativistic cases [see A.O. Barut (1965) or J.L. Anderson (1967)] with some subtleties concerning the constraint uµuµ = 1 in the latter case [G. Kalman (1961); A. Peres, N. Rosen (1960)].
In the same way, the equations satisfied by the fields (continuous systems with an infinite number of degrees of freedom), whatever tensor nature they may have, can often be deduced from variational principles. This is true of the equations of electromagnetism, for example, but not the equation for heat transfer.
There are many analytic procedures for describing the motion of a particle or the evolution of a field. There is no a priori reason to confine oneself to differential equations or second order partial differential equations.
The main advantage of a variational formalism is that it allows one to find the conserved quantities in the motion (i.e. the first integrals) and directly to exploit the symmetries of the physical problem considered; these two aspects are connected, as we shall see. We shall introduce such a formalism here only because it allows us to define the energy and momentum of a field very simply.
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- Information
- An Introduction to Relativistic Gravitation , pp. 245 - 248Publisher: Cambridge University PressPrint publication year: 1999